Miniaturized device to sterilize surfaces from Covid-19 and other viruses and bacteria

ABSTRACT

A system for sterilizing biological material comprising beam generation circuitry for generating a radiating wave having radiating energy therein at a predetermined frequency therein. A controller controls the radiating wave generation at the predetermined frequency. The predetermined frequency equals a resonance frequency of a particular biological material and is determined responsive to a plurality of parameters from an influenza virus. The predetermined frequency induces a mechanical resonance vibration at the resonance frequency of the particular biological material within the particular biological material for destroying a capsid of the particular biological material. Radiating circuitry projects the radiating wave on a predetermined location to destroy the particular biological material at the predetermined location.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 16/925,107, entitled A MINIATURIZED DEVICE TO STERILIZE FROMCOVID-19 AND OTHER VIRUSES, filed on Jul. 9, 2020. U.S. patentapplication Ser. No. 16/925,107 is a continuation-in-part of U.S. patentapplication Ser. No. 16/127,729, entitled SYSTEM AND METHOD FOR APPLYINGORTHOGONAL LIMITATIONS TO LIGHT BEAMS USING MICROELECTROMECHANICALSYSTEMS, filed on Sep. 11, 2018, which is incorporated herein byreference. U.S. patent application Ser. No. 16/925,107 also acontinuation-in-part of U.S. patent application Ser. No. 16/653,213,entitled SYSTEM AND METHOD FOR MULTI-PARAMETER SPECTROSCOPY, filed onOct. 15, 2019, which is incorporated herein by reference. U.S. patentapplication Ser. No. 16/925,107 claims benefit of U.S. ProvisionalPatent Application No. 63/032,256, entitled A MINIATURIZED DEVICE TOSTERILIZE SURFACES FROM COVID-19 AND OTHER VIRUSES, filed on May 29,2020, which is incorporated herein by reference.

This application also claims priority to U.S. Provisional PatentApplication No. 63/074,298, entitled A MINIATURIZED DEVICE TO STERILIZESURFACES FROM COVID-19 AND OTHER VIRUSES AND BACTERIA, filed on Sep. 3,2020, which is incorporated herein by reference.

TECHNICAL FIELD

The present invention relates to the detection and sterilization ofviruses, and more particular to the detection and sterilization of virususing orbital angular momentum.

BACKGROUND

The spread of viruses presents a challenge to protecting individuals ina society where people live in close proximity to each other andcommonly use areas in restaurants, offices, hotels and other public usefacilities. The greatest challenge in these types of facilities is thesanitation of surfaces that people come in common contact with in thesepublic and common use facilities. Current techniques involve the use ofdisinfectants to wipe down the commonly used surfaces and chemicallykill the viruses or other biological materials on the surfaces. However,the use of disinfectants that must be wiped onto a surface can sometimesresult in an incomplete disinfection since the entire surface must betouched in the physical cleaning of the surface. Additionally, areasother than surfaces must be sterilized from viruses. Thus, the abilityto more completely cover the entirety of a surface or other area duringthe disinfection process could great help in limiting the spread ofviruses or other contaminants that may be spread from contact withcontaminated surfaces.

SUMMARY

The present invention, as disclosed and described herein in one aspectthereof, comprises a system for sterilizing biological materialcomprising beam generation circuitry for generating a radiating wavehaving radiating energy therein at a predetermined frequency therein. Acontroller controls the radiating wave generation at the predeterminedfrequency. The predetermined frequency equals a resonance frequency of aparticular biological material and is determined responsive to aplurality of parameters from an influenza virus. The predeterminedfrequency induces a mechanical resonance vibration at the resonancefrequency of the particular biological material within the particularbiological material for destroying a capsid of the particular biologicalmaterial. Radiating circuitry projects the radiating wave on apredetermined location to destroy the particular biological material atthe predetermined location.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding, reference is now made to thefollowing description taken in conjunction with the accompanyingDrawings in which:

FIG. 1A illustrates the disabling of viruses using Eigen vibrations;

FIG. 1B illustrates a system for disabling viruses;

FIG. 2 illustrates the combination of charges and microwave energy tocreate mechanical oscillations and viruses;

FIG. 3 illustrates a Covid-19 virus;

FIG. 4 illustrates the geometry of Casper-Klug classification ofviruses;

FIGS. 5A-C illustrates charge distribution on the capsid of a virus;

FIG. 6 illustrates a flow diagram of a process for providing energy to avirus;

FIGS. 7A-D illustrates different modes of Hermite-Gaussian forapplication to a virus;

FIG. 8 illustrates the optoacoustical generation of a helicoidalultrasonic beam;

FIG. 9 illustrates various manners for generating a beam having orbitalangular momentum applied thereto;

FIG. 10 illustrates the application of an OAM beam to a virus;

FIG. 11 illustrates the application of various OAM modes to a virus;

FIGS. 12A-B illustrates the application of different Hermite Gaussianmode signals to a virus;

FIG. 13 illustrates a patch antenna;

FIG. 14 illustrates patch antennas for providing different OAM modes;

FIG. 15 illustrates different intensity signatures provided from varyingdistances;

FIG. 16 is a block diagram of circuitry for applying signals to a virusto create mechanical resonance therein;

FIG. 17 illustrates a sterilization system implemented within a cellphone;

FIG. 18 illustrates a sterilization system implemented within a handheldflashlight;

FIG. 19 illustrates a sterilization system implemented within a portablelight wand;

FIG. 20 illustrates a sterilization system implemented within aflorescent light mounted on a ceiling;

FIG. 21 illustrates a sterilization system implemented within aflorescent light mounted on a wall;

FIG. 22 illustrates a sterilization system implemented within anincandescent bulb.

FIG. 23 illustrates interaction between OAM light and graphene;

FIG. 24 illustrates a graphene lattice in a Honeycomb structure;

FIG. 25 illustrates the components of a vector beam;

FIG. 26 illustrates the use of electromagnetic waves for the transfer ofenergy to a virus;

FIG. 27 illustrates the use of orbital angular momentum for the transferof energy to a virus;

FIG. 28 is a functional block diagram of a system for generating orbitalangular momentum within a communication system;

FIG. 29 illustrates a single wavelength having two quanti-spinpolarizations providing an infinite number of signals having variousorbital angular momentums associated therewith;

FIG. 30A illustrates an object with only a spin angular momentum;

FIG. 30B illustrates an object with an orbital angular momentum;

FIG. 30C illustrates a circularly polarized beam carrying spin angularmomentum;

FIG. 30D illustrates the phase structure of a light beam carrying anorbital angular momentum;

FIG. 31 illustrates a light beam having orbital angular momentumimparted thereto;

FIG. 32 illustrates a series of parallel wavefronts;

FIG. 33 illustrates a wavefront having a Poynting vector spiralingaround a direction of propagation of the wavefront;

FIG. 34 illustrates a plane wavefront;

FIG. 35 illustrates a helical wavefront;

FIG. 36 illustrates a plane wave having only variations in the spinvector;

FIG. 37 illustrates the application of a unique orbital angular momentumto a wave;

FIGS. 38A-38C illustrate the differences between signals havingdifferent orbital angular momentum applied thereto;

FIG. 39A illustrates the propagation of Poynting vectors for variouseigenmodes;

FIG. 39B illustrates a spiral phase plate;

FIG. 40 illustrates a block diagram of an apparatus for providingconcentration measurements and presence detection of various materialsusing orbital angular momentum;

FIG. 41 illustrates an emitter of the system of FIG. 40 ;

FIG. 42 illustrates a fixed orbital angular momentum generator of thesystem of FIG. 40 ;

FIG. 43 illustrates one example of a hologram for use in applying anorbital angular momentum to a plane wave signal;

FIG. 44 illustrates the relationship between Hermite-Gaussian modes andLaguerre-Gaussian modes;

FIG. 45 illustrates super-imposed holograms for applying orbital angularmomentum to a signal;

FIG. 46 illustrates a tunable orbital angular momentum generator for usein the system of FIG. 11 ;

FIG. 47 illustrates a block diagram of a tunable orbital angularmomentum generator including multiple hologram images therein;

FIG. 48 illustrates the manner in which the output of the OAM generatormay be varied by applying different orbital angular momentums thereto;

FIG. 49 illustrates an alternative manner in which the OAM generator mayconvert a Hermite-Gaussian beam to a Laguerre-Gaussian beam;

FIG. 50 illustrates the manner in which holograms within an OAMgenerator may twist a beam of light;

FIG. 51 illustrates the manner in which a sample receives an OAM twistedwave and provides an output wave having a particular OAM signature;

FIG. 52 illustrates the manner in which orbital angular momentuminteracts with a molecule around its beam axis;

FIG. 53 illustrates a block diagram of the matching circuitry foramplifying a received orbital angular momentum signal;

FIG. 54 illustrates the manner in which the matching module may usenon-linear crystals in order to generate a higher order orbital angularmomentum light beam;

FIG. 55 illustrates a block diagram of an orbital angular momentumdetector and user interface;

FIG. 56 illustrates the effect of sample concentrations upon the spinangular polarization and orbital angular polarization of a light beampassing through a sample;

FIG. 57 more particularly illustrates the process that alters theorbital angular momentum polarization of a light beam passing through asample;

FIG. 58 provides a block diagram of a user interface of the system ofFIG. 12 ;

FIG. 59 provides a block diagram of a more particular embodiment of anapparatus for measuring the concentration and presence of glucose usingorbital angular momentum;

FIG. 60 is a flow diagram illustrating a process for analyzing intensityimages;

FIG. 61 illustrates an ellipse fitting algorithm;

FIG. 62 illustrates the generation of fractional orthogonal states;

FIG. 63 illustrates the use of a spatial light modulator for thegeneration of fractional OAM beams;

FIG. 64 illustrates one manner for the generation of fractional OAM beamusing superimposed Laguerre Gaussian beams;

FIG. 65 illustrates the decomposition of a fractional OAM beam intointeger OAM states;

FIG. 66 illustrates the manner in which a spatial light modulator maygenerate a hologram for providing fractional OAM beams;

FIG. 67 illustrates the generation of a hologram to produce non-integerOAM beams;

FIG. 68 is a flow diagram illustrating the generation of a hologram forproducing non-integer OAM beams;

FIG. 69 is a block diagram illustrating fractional OAM beams for OAMspectroscopy analysis;

FIG. 70 illustrates an example of an OAM state profile;

FIG. 71 illustrates the manner for combining multiple variedspectroscopy techniques to provide multiparameter spectroscopy analysis;

FIG. 72 illustrates a schematic drawing of a spec parameter for makingrelative measurements in an optical spectrum;

FIG. 73 illustrates the stretching and bending vibrational modes ofwater;

FIG. 74 illustrates the stretching and bending vibrational modes forCO₂;

FIG. 75 illustrates the energy of an anharmonic oscillator as a functionof the interatomic distance;

FIG. 76 illustrates the energy curve for a vibrating spring andquantized energy level;

FIG. 77 illustrates Rayleigh scattering and Ramen scattering by Stokesand anti-Stokes resonance;

FIG. 78 illustrates circuits for carrying out polarized Rahmantechniques;

FIG. 79 illustrates circuitry for combining polarized and non-polarizedRahman spectroscopy;

FIG. 80 illustrates a combination of polarized and non-polarized Rahmanspectroscopy with optical vortices;

FIG. 81 illustrates the absorption and emission sequences associatedwith fluorescence spectroscopy;

FIG. 82 illustrates a combination of OAM spectroscopy with Ramenspectroscopy for the generation of differential signals;

FIG. 83 illustrates a flow diagram of an alignment procedure

FIG. 84 illustrates a top view of a multilayer patch antenna array;

FIG. 85 illustrates a side view of a multilayer patch antenna array;

FIG. 86 illustrates a first layer of a multilayer patch antenna array;

FIG. 87 illustrates a second layer of a multilayer patch antenna array;

FIG. 88 illustrates a transmitter for use with a multilayer patchantenna array;

FIG. 89 illustrates a microstrip patch antenna;

FIG. 90 illustrates a coordinate system for an aperture of a microstrippatch antenna;

FIG. 91 illustrates a 3-D model of a single rectangular patch antenna;

FIG. 92 illustrates the radiation pattern of the patch antenna;

FIG. 93 illustrates microwave resonance absorption measurements;

FIG. 94 illustrates attenuation spectra and background attenuationspectra;

FIG. 95 illustrates a plot of resonance frequency;

FIG. 96 illustrates maximum transmit power to achieve IEEE safety powerdensity thresholds for a first theta;

FIG. 97 illustrates maximum transmit power to achieve IEEE safety powerdensity thresholds for a second theta;

FIG. 98 illustrates maximum transmit power to achieve IEEE safety powerdensity thresholds for a third theta;

FIG. 99 a illustrates a plot of dE/dw versus distance;

FIG. 99 b illustrates a further plot of dE/dw versus distance;

FIG. 100 illustrates a plot of a minimum of electric field as a functionof distance; and

FIG. 101 illustrates various manners for implementing an antenna fortransmitting signals to generate a residence within a virus or bacteria.

DETAILED DESCRIPTION

Referring now to the drawings, wherein like reference numbers are usedherein to designate like elements throughout, the various views andembodiments of a miniaturized device to sterilize from COVID-19 andother viruses are illustrated and described, and other possibleembodiments are described. The figures are not necessarily drawn toscale, and in some instances the drawings have been exaggerated and/orsimplified in places for illustrative purposes only. One of ordinaryskill in the art will appreciate the many possible applications andvariations based on the following examples of possible embodiments.

How quickly will our citizens revert to their old lifestyle patterns asour country begins to relax coronavirus related shelter-in-placeguidelines? The economic wellbeing of our country turns on the answer tothis question. A technology is proposed that can be embedded intodifferent devices for the purpose of sterilizing Coronavirus (COVID-19)and other viruses and bacteria from surfaces as well as airborne spaces.If this technology is embedded into a handheld device (size of a smallflashlight) or into a cell phone handset, then most of the populationcould use it to stop or minimize the propagation of the virus. Thetechnology can also be embedded into fixed light fixtures to sanitizespaces without the use of harmful ionizing radiation (i.e. UVC). Thougha lot of genetic information is already available about COVID-19, itsphysical properties are largely unknown. Identifying the physicalvirology information of this virus would allow us to develop safe andeasy to use products that would sanitize many different environments.The required information includes: the frequency of a non-ionizingelectromagnetic radiation (i.e. microwave region), the frequency ofmechanical resonance, as well as stress levels needed to rupture thecapsid of the virus.

The rational is that electromagnetic radiation can destroy the virus,but this radiation needs to be in a form that is entirely safe fordirect human use and therefore non-ionizing. Microwave radiation caninduce plasma oscillations on charge distribution of the virus, therebycreating mechanical (ultrasonic) longitudinal eigen-vibrations torupture the capsid of the virus. In addition, this electromagneticradiation can be a structured vector beam with Laguerre-Gaussian orHermite Gaussian intensity so that transverse shear forces or torsion becan be imparted to the virus's icosahedral lattice structure where thefrequencies are safe for humans and non-ionizing.

A theoretical model based on the size, geometry, and protein material ofthe virus enables identification of a range of electromagneticfrequencies needed to induce plasma oscillations on the chargedistribution of the virus. The identification of the dominant frequencywithin the theoretical spectrum reduces trial and error experimentsgiven the theoretical model. Aim 3 is to identify the theoretical andexperimental mechanical stresses needed to rupture the capsid of thevirus are identified and the frequency of the mechanicaleigen-vibrations to achieve the required stresses are determined.Finally, the electromagnetic intensity required for the radiation isdetermined where the frequency of the radiation would match theeigen-vibrations and the intensity of the radiation would be slightlyhigher than the stresses needed to ensure the capsid will rupture

The described technology will overcome the safety concerns regarding theuse of ionizing radiations and provide an approach that will give ourcitizens the confidence to return to their normal routines with a simplemethod to sterilize surfaces and spaces from COVID-19. It also gives agreater insight into the physical virology of coronavirus, knowledgethat is currently lacking and that promises to yield novel insights intovirology and molecular biology. This new technology will provide avaluable resource even with respect to other viruses and bacteria withperhaps different frequencies, intensities, and modes of its structuredvector beam.

A miniaturized device that can be embedded into a handheld unit (size ofa small flashlight) or into a cell phone handset for sterilizingsurfaces from Coronavirus (COVID-19) or other viruses/biologicalmaterials that radiates similar to flashlight on the cell phone but at adifferent frequency would provide a great benefit to heavy public useareas that require constant cleaning.

Inducement of Vibrations to COVID-19 and Other Viruses and Bacteria

The below described techniques for the generation Laguerre-Gaussian,Hermite-Gaussian, or Ince-Gaussian processed beams provide an improvedmanner for the sterilization from COVID-19 or other viruses andbacteria. The above described techniques may be used for generating aLaguerre-Gaussian, Hermite-Gaussian, or Ince-Gaussian beam that impartsresonance vibrations to the structures of COVID-19 or other viruses andbacteria in order to destroy on inactivate the virus. The circuitry forinducing the resonance may be provided in a handheld portable devicesimilar to a flashlight or light wand or within a cell phone. Also, thecircuitry can be implemented within a normal lighting fixture.

The following describes a miniaturized device that can be embedded intoa handheld unit (size of a small flashlight) or into a cell phonehandset for the purpose of sterilizing surfaces or areas fromCoronavirus (COVID-19) or other viruses and bacteria. The objective ofthe device is to either kill or disable the virus. The product conceptcan easily extend to devices that can be plugged into the connectors ofa regular lamp or fluorescent light for fixed applications. Surfaceareas can be illuminated for sterilizing objects as well as volumes thatmay contain airborne viruses.

This system also describes how energy configurations transmitted usingantennas such as patch antenna arrays, horn and conical antennas can beused to sterilize surfaces or areas by transmitting signals at a givenfrequency. The system utilizes photonic and ultrasonic sources thatcould kill the viruses. As shown in FIG. 1A because viruses 102 can bedisabled by inducing specific eigen vibrations 104 using safe microwavesand/or ultrasonic energy 106, this approach is quite different thanusing ionizing radiation (i.e. alpha, beta, gamma or even x-rays andultraviolet). In this approach, the viruses 102 are killed by leveraginga natural sensitivity of the virus to certain resonant frequencies witha vector beam 106 specifically engineered to induce the frequency withinthe virus and kill the virus based on its size, structure, geometry,material (proteins) and boundary conditions. These vector beams 106could take the form of Laguerre-Gaussian, Hermite-Gaussian, orInce-Gaussian in both electromagnetic as well as ultrasonic waves toinduce torsional, shear and longitudinal vibrations 104 to rupture thecapsid of the virus 102. These beams can also be manually focused toincrease the power density of the field for sterilizing keys, doorhandles or other objects.

Referring now to FIG. 1B, there is illustrated a general block diagramof a system for disabling a virus as discussed in FIG. 1A. Theparticular details of the system will be more fully described hereinbelow. A system 110 for sterilizing viruses includes beam generationcircuitry 114. The beam generation circuitry 114 for generates aradiating wave beam 116 having radiating energy therein at apredetermined frequency therein. A controller 118 controls the radiatingwave beam to be generated at the predetermined frequency. Thepredetermined frequency equals a resonance frequency of a specific virus120. The predetermined frequency induces a mechanical resonancevibration at the resonance frequency of the specific virus within thevirus 120. The induced mechanical resonance vibration destroys a capsidof the particular virus 120 and destroys the virus. Radiating circuitry122 projects the radiating wave on a predetermined location to destroythe particular virus 120 at the predetermined location.

Introduction

Three models have been developed for mechanical resonance of differentgeometries as described in S. Ashrafi, et al. “Spurious Resonances andModeling of Composite Resonators,” IEEE Proceedings of the 37th AnnualSymposium on Frequency Control, 1983 which is incorporated herein byreference. The first model was a one-dimensional model that couldpredict the principal resonances of a given geometry but failed toaccount for spurious resonances which were observed experimentally. Thetwo-dimensional model showed a refinement to predict the qualitativestructure of the spectrum. However, a three-dimensional model not onlypredicted the dominant resonances, but it also predicted the spuriousresonances of the geometry.

A model was also developed to predict the eigen-vibrations of an elasticbody which could be traced to the excitation of circumferential waves inS. Ashrafi, et al. “Acoustically Induced Stresses in Elastic Cylindersand Their Visualization,” J. Acoust. Soc. Am. 82 (4), October 1987 whichis incorporated herein by reference. These waves propagate along thesurface of the body, inside the body material, and partly also inambient medium. In fact, an energy transfer from acoustic, ultrasonic,or mechanical waves to electromagnetic birefringence was shown. Thesebirefringence patterns are different for different geometries (i.e.cylindrical, spherical, etc.)

A new property of photons related to electromagnetic (EM) vortices thatcarry orbital angular momentum (OAM) have been leveraged to detectcertain molecules or tumors and also use such vectors beams to destroyor break up the molecules as described in A. Siber, et al. “Energies andpressures in viruses: contribution of nonspecific electrostaticinteractions,” Phys. Chem. Chem. Phys., 2012, 14, 3746-3765; A. L.Bozic, et al. “How simple can a model of an empty viral capsid be?Charge distributions in viral capsids,” J Biol Phys. 2012 September;38(4): 657-671; S. Ashrafi, et al. “Recent advances in high-capacityfree-space optical and radio-frequency communications using orbitalangular momentum multiplexing,” Royal Society Publishing, Phil. Trans.R. Soc. A375:20150439, Oct. 13, 2016; S. Ashrafi, et al. “PerformanceMetrics and Design Parameters for an FSO Communications Link Based onMultiplexing of Multiple Orbital-Angular-Momentum Beams,” Globecom2014OWC Workshop, 2014; S. Ashrafi, et al. “Optical Communications UsingOrbital Angular Momentum Beams,” Adv. Opt. Photon. 7, 66-106, Advancesin Optics and Photonic, 2015; S. Ashrafi, et al. “PerformanceEnhancement of an Orbital-Angular-Momentum-Based Free-Space OpticalCommunication Link through Beam Divergence Controlling,” OSA, 2015; S.Ashrafi, et al. “Link Analysis of Using Hermite-Gaussian Modes forTransmitting Multiple Channels in a Free-Space Optical CommunicationSystem,” The Optical Society, Vol. 2, No. 4, April 2015; S. Ashrafi, etal. “Performance Metrics and Design Considerations for a Free-SpaceOptical Orbital-Angular-Momentum-Multiplexed Communication Link,” OSA,Vol. 2, No. 4, April 2015; S. Ashrafi, et al. “Demonstration of DistanceEmulation for an Orbital-Angular-Momentum Beam,” OSA, 2015; S. Ashrafi,et al. “Free-Space Optical Communications Using Orbital-Angular-MomentumMultiplexing Combined with MIMO-Based Spatial Multiplexing,” OpticsLetters, 2015, each of which are incorporated herein by reference.

Vector beams have been used for advanced spectroscopy with specificinteraction signatures with matter as described in S. Ashrafi, et al.“Orbital and Angular Momentum Multiplexed Free Space OpticalCommunication Link Using Transmitter Lenses,” Applied Optics, Vol. 55,No. 8, March 2016; S. Ashrafi, et al. “Experimental Characterization ofa 400 GBit/s Orbital Angular Momentum Multiplexed Free Space OpticalLink Over 120 m,” Optics Letters, 2016, which are incorporated herein byreference. We further studied OAM light-matter interactions using suchvector beams as well as photon-phonon interaction with detailedinteraction Hamiltonians have also been studied as described in S.Ashrafi, et al. “Orbital and Angular Momentum Multiplexed Free SpaceOptical Communication Link Using Transmitter Lenses,” Applied Optics,Vol. 55, No. 8, March 2016; S. Ashrafi, et al. “ExperimentalCharacterization of a 400 GBit/s Orbital Angular Momentum MultiplexedFree Space Optical Link Over 120 m,” Optics Letters, 2016; S. Ashrafi,et al. “Demonstration of OAM-based MIMO FSO link using spatial diversityand MIMO equalization for turbulence mitigation,” Optical Fiber Conf;OSA 2016, which are incorporated herein by reference.

Though earlier work has covered the transfer of energy from acoustic orultrasonic waves to electromagnetic birefringence, the current systemdescribes a reverse of the earlier work so that electromagnetic wavesare incident on specific geometries (i.e. viruses) 102 and mechanicaleigen-vibrations 104 are induced on the virus to destroy it 108. Ourradiated vector beams 106 can also carry OAM to impart mechanical torqueto the virus structure as well as possible UVC at powers that are safefor short periods of time.

Methodology

The methodology used in this system leverages previous published papersand patents and add new approaches and unique ingredients forsterilizing surfaces, volumes, and objects from COVID-19 or otherviruses and bacteria. The uses the techniques described in the abovementioned S. Ashrafi, et al. “Spurious Resonances and Modeling ofComposite Resonators,” IEEE Proceedings of the 37^(th) Annual Symposiumon Frequency Control, 1983 and S. Ashrafi, et al. “Acoustically InducedStresses in Elastic Cylinders and Their Visualization,” J. Acoust. Soc.Am. 82 (4), October 1987. The methodology also uses techniques involvingstructured vector beams with Laguerre-Gaussian (LG), Hermite-Gaussian(HG), Ince-Gaussian (IG) as well as orthogonal Spheroidal structures forboth electromagnetic and ultrasonic waves. On the electromagnetic beamsthe systems include frequencies from radio frequencies to highermillimeter waves to microwaves all the way to infrared (IR), visiblelight and ultraviolet (UV); techniques on vector beams with LG structurethat carry Orbital Angular Momentum (OAM) or Fractional OAM with IR thatuses absorption due to vibration (changes of dipolemoment/Polarization); techniques on vector beams with LG structure thatcarry Orbital Angular Momentum (OAM) or Fractional OAM with Rayleigh andRaman spectroscopy; techniques on vector beams with LG structure thatcarry Orbital Angular Momentum (OAM) or Fractional OAM with IRspectroscopy that can have certain vibrational modes forbidden in IRwith better signal/noise ratio; techniques on vector beams with LGstructure that carry Orbital Angular Momentum (OAM) or Fractional OAMwith Spontaneous, Stimulated, Resonance and Polarized Raman; techniqueson vector beams with LG structure that carry Orbital Angular Momentum(OAM) or Fractional OAM with Tera Hertz (THz) Spectroscopy; techniqueson vector beams with LG structure that carry Orbital Angular Momentum(OAM) or Fractional OAM with Fluorescence Spectroscopy; techniques onvector beams with LG structure that carry Orbital Angular Momentum (OAM)or Fractional OAM with pump & probe for Ultrafast Spectroscopy;techniques on vector beams with LG structure that carry Orbital AngularMomentum (OAM) or Fractional OAM with pump & probe for UltrafastSpectroscopy; techniques on detection, tomography and destruction oftumors using structured beams; techniques on focusing structured vectorbeams; techniques on application of horn, conical and patch antennas forcreation of structured beams; techniques on photon-phonon interactions;techniques on light-matter interaction; techniques on interactionHamiltonians for quantum dots (This may allow finding of a method formaking quantum dots of any size to either couple the virus to a quantumdot or encapsulating a fluorescent quantum dot inside a virus which willallow scientists a better understand physical virology and new ways toprevent viral infection.); techniques on Interaction Hamiltonian of LGbeams with Graphene lattice. The honeycomb lattice of Graphene is ahexagonal lattice with primitive vectors that very much look like theCasper-Klug (CK) vectors for describing the hexamer structure forcapsomers on the capsid of the virus. This enables building of asynthetic capsid lattice out of graphene for multiple purposes.

Approach

Mechanical properties of viruses have been studied experimentally usingAtomic Force and Electron Microscopy (AFM). Also, mechanical, elastic,and electrostatic properties of viruses have been studied by severalscientists. It was observed that under certain physiological conditions,virus capsid assembly requires the presence of genomic material that isoppositely charged to the core proteins. There is also some work on thepossibility of inducing photon-phonon interactions in viruses.

Referring now to FIG. 2 , the resonance phenomenon of the viruses is dueto separation of positive-negative electric charges 102 on the body ofthe virus particles and the coupling of microwave energy 104 through theinteraction with the three dimensional bipolar electric chargesdistributions, generating mechanical oscillations 106 at the samefrequency. At specific microwave frequencies depending on the diameterand other properties of the virus, primarily the dipole acoustic mode,can be purposed as a mechanism to induce eigen-vibrations to viruses andkill them. The phenomenon here is of non-thermal nature related tonon-ionizing radiation. Raman scattering phenomena should be able toshow the existence of acoustic-mechanical resonance phenomena inviruses.

For the Covid-19 global pandemic, it is difficult to open the societyfrom quarantine unless there is a way to sterilize spaces such as publicvenues, hospitals, clinics, commercial buildings, restaurants . . . etc.This can be done by large devices for big commercial use as well assmall handheld unit (size of a small flashlight) or embedded into a cellphone handset for consumer use.

Current airborne virus epidemic prevention used in public space includesstrong chemicals, UV irradiation, and microwave thermal heating. Allthese methods affect the open public. However, we know that ultrasonicenergy can be absorbed by viruses. Viruses can be inactivated bygenerating the corresponding resonance ultrasound vibrations of viruses(in the GHz). Several groups started investigating the vibrational modesof viruses in this frequency range. The dipolar mode of the acousticvibrations inside viruses can be resonantly excited by microwaves of thesame frequency with a resonant microwave absorption effect. Theresonance absorption is due to an energy transfer from electromagneticwaves to acoustic vibration of viruses. This is an efficient way toexcite the vibrational mode of the whole virus structure because of a100% energy conversion of a photon into a phonon of the same frequency,but the overall efficiency is also related to the mechanical propertiesof the surrounding environment. We would like the energy transfer frommicrowave to virus vibration be just enough to kill the virus whilemicrowave power density be safe in open public places.

Induced stress (vibrations) on the virus can fracture its structure andthe microwave energy needed is to achieve the virus inactivationthreshold. These thresholds can be identified for different viruses atdifferent microwave power densities.

Higher inactivation of viruses can be achieved at the dipolar resonantfrequency. It is also important that at the resonant frequency, themicrowave power density threshold for virus inactivation is below theIEEE safety standard. The main inactivation mechanism is throughphysically fracturing the viruses while the RNA genome is not degradedby the microwave illumination, supporting the fact that this approach isfundamentally different from the microwave thermal heating effect.

Framework

Referring now to FIG. 3 , COVID-19 is a spherical shape virus withdiameter of 100 nm. Since the protein and genome have similar mechanicalproperties, for the estimation of dipolar vibration frequencies, thevirus can be treated as a homogenous sphere.

The virus has four structural proteins, known as the S (spike) 304, E(envelope) 306, M (membrane) 308, and N (nucleocapsid) proteins 310. TheN protein 310 holds the RNA genome, and the S protein 304, E protein306, and M protein 308 together create the viral envelope. The spikeprotein 304, which has been imaged at the atomic level using cryogenicelectron microscopy, is the protein responsible for allowing the virusto attach to and fuse with the membrane of a host cell. The envelope,composed mainly of lipids 312, which can be destroyed with alcohol orsoap

There is quantization of certain physical parameters in both quantummechanics as well as classical mechanics due to boundary conditions.Such quantization is seen in quantum dots and nanowires. In 1882, Lambstudied the torsional and spheroidal modes of a homogeneous sphere byconsidering thestress-free boundary condition on the surface. Amongthese modes, the SPH mode with

=1 allows dipolar coupling and the corresponding eigenvalue equation canbe expressed as:

${{4\frac{j_{2}(\xi)}{j_{1}(\xi)}\xi} - \eta^{2} + {2\frac{j_{2}(\eta)}{j_{1}(\eta)}\eta}} = 0$where$\xi = {{\frac{\omega_{0}R}{V_{L}}\mspace{14mu}\eta} = \frac{\omega_{0}R}{V_{T}}}$j_(t)=spherical Bessel functionω₀=angular frequency of vibrational modeR=Radius of virus (i.e. 100 nm)V_(L)=longitudinal mechanical velocityV_(T)=Transverse mechanical velocity

=0 breathing mode

=1 dipolar mode

=2 quadrupole mode

When a resonantly oscillating electric field is applied to thenano-sphere, opposite displacement between core and shell can begenerated and further excite the dipolar mode vibrations. Dipole mode isthe only spherical mode to directly interact with the EM waves withwavelength much longer than the virus size. Due to the permanent chargeseparation nature of viruses, dipolar coupling is the mechanismsresponsible for microwave resonant absorption in viruses by treatingspherical viruses as homogeneous nanoparticles.

Electromagnetic-Mechanical Lorentz-Type Model

Let's describe the virus as a sphere with charge distribution and applya damped mass-spring model.m _(*) {umlaut over (x)}+m _(*) γ{dot over (x)}+kx=0 m _(*)=effectivemassIf damping=0Thenx(t)=A ₀ sin(ω₀ t)+B ₀ cos(ω₀ t)Or

${x(t)} = {{X_{0}\mspace{14mu}{\cos\left( {{\omega\; t} - \phi} \right)}\mspace{14mu}\omega_{0}} = \sqrt{\frac{k}{m}}}$

Now when we put this virus in an external {right arrow over (E)}electric field of microwave frequency, we havem _(*) {umlaut over (x)}+m _(*) γ{dot over (x)}+kx=qE E=E ₀ cos(ωt)

Using Laplace transform, the solution would bex(t)=X ₀ cos(ωt−ϕ)Where

$X_{0} = {{\frac{{qE}_{0}}{m_{*}\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2} - \left( \frac{\omega_{0}\omega}{Q} \right)^{2}}}\mspace{14mu}{and}\mspace{14mu}\tan\mspace{14mu}\phi} = \frac{\omega_{0}\omega}{Q\left( {\omega_{0}^{2} - \omega^{2}} \right)}}$

Because of damping the decay rate of oscillation is equal to theimaginary part of the frequency

$\frac{\omega_{0}}{2Q} = {{\frac{m_{*}\gamma}{2m_{*}}\mspace{14mu} m_{*}\gamma} = \frac{\omega_{0}m_{*}}{Q}}$

Ifx(t)=X ₀ e ^(iωt){dot over (x)}(t)=iωX ₀ e ^(iωt){circumflex over (x)}(t)=(iω)² X ₀ e ^(iωt)=−ω² x(t)Therefore

${{m_{*}\overset{¨}{x}} + {m_{*}\gamma\;\overset{.}{x}} + {kx}} = {{{q\overset{\rightarrow}{E}} - {m_{*}\omega^{2}{x(t)}} + {m_{*}{\gamma\left( {i\;\omega} \right)}{x(t)}} + {{kx}(t)}} = {{{q\overset{\rightarrow}{E}} - {m_{*}\omega^{2}x} + {{im}_{*}{\omega\gamma}\; x} + {kx}} = {{{qE}\mspace{14mu}{x(t)}} = {X_{0}\mspace{14mu}{\cos\left( {{\omega\; t} + \phi} \right)}}}}}$$\omega = \frac{{{im}_{*}\gamma} \pm \sqrt{{- \left( {m_{*}\gamma} \right)^{2}} + {4{km}_{*}}}}{2m_{*}}$

The power absorption of thisP _(abs) =qEv−qE ₀ cos(ωt)X ₀ω sin(ωt−ϕ)

Therefore, for one full cycle we have

$\left\langle P_{abs} \right\rangle = {{\frac{1}{2}\frac{{Q\left( {qE}_{0} \right)}^{2}\omega_{0}\omega^{2}}{{Q^{2}{m_{*}\left( {\omega_{0}^{2} - \omega^{2}} \right)}^{2}} + {\left( {\omega_{0}\omega} \right)^{2}m_{*}}}} = \frac{\omega_{0}\omega^{2}m_{*}X_{0}^{2}}{2Q}}$

The absorption cross-section of the virus

$\sigma_{abs} = \frac{\left\langle P_{abs} \right\rangle}{powerflux}$

Let's try to breakup the outer layer (may be the lipid membrane of theenvelope protein)

Estimate the Max Induced Stress

${stress}_{\max} = \frac{F_{\max}^{induced}}{{Area}\mspace{14mu}{of}\mspace{14mu}{shell}}$

Let's assume maximum induced stress=α average and the shell covers β% ofequatorial plane

${stress}_{\max} = {\frac{\alpha\;{kX}_{0}}{{\beta\pi}\; r^{2}} = {{\frac{\alpha}{\beta}\frac{m_{*}\omega_{0}^{2}X_{0}}{\pi\; r^{2}}} = {\frac{\alpha}{\beta}\frac{m_{*}\omega_{0}^{2}}{\pi\; r^{2}}\frac{{qE}_{0}}{\sqrt{{m_{*}^{2}\left( {\omega_{0} - \omega^{2}} \right)}^{2} + {\left( \frac{\omega_{0}m_{*}}{Q} \right)^{2}\omega^{2}}}}}}}$

Therefore, if the stress is known experimentally from UTSW then theintensity of microwave can be found to be

$E_{0} = {\left( {stress}_{\max} \right)\frac{\beta}{\alpha}\frac{\pi\; r^{2}\sqrt{{m_{*}^{2}\left( {\omega_{0}^{2} - \omega^{2}} \right)}^{2} + {\left( \frac{\omega_{0}m_{*}}{Q} \right)^{2}\omega^{2}}}}{{qm}_{*}\omega_{0}^{2}}}$${{m_{*}\overset{¨}{x}} + {m_{*}\gamma\;\overset{.}{x}} + {m_{*}\omega_{0}^{2}x}} = {q\overset{\rightarrow}{E}}$${\overset{¨}{x} + {\gamma\;\overset{.}{x}} + {\omega_{0}^{2}x}} = {\frac{q}{m_{*}}E}$Eαe^(iωt) time variation

${{{- \omega^{2}}x} + {j\;{\omega\gamma}\; x} + {\omega_{0}^{2}x}} = {{\frac{q}{m_{*}}E\mspace{14mu}{x\left\lbrack {{- \omega^{2}} + {j\;{\omega\gamma}} + \omega_{0}^{2}} \right\rbrack}} = {\frac{q}{m_{*}}E}}$$x = {\frac{q}{m_{*}}\frac{1}{\left( {\omega_{0}^{2} - \omega^{2}} \right) + {j\;{\omega\gamma}}}E}$

Since charge displacement x(t)α polarization {right arrow over (P)}

$\overset{\rightarrow}{D} = {{{\epsilon_{0}\overset{\rightarrow}{E}} + {\overset{\rightarrow}{P}\mspace{14mu} P_{x}}} = {{{{Nqx}\left( {\frac{d}{{dt}^{2}} + {\gamma\frac{d}{dt}} + \omega_{0}^{2}} \right)}{P_{x}(t)}} = {{\frac{{Nq}^{2}}{m_{*}}{E(t)}} = {\epsilon_{0}\omega_{p}^{2}{E(t)}}}}}$$\omega_{p}^{2} = \frac{{Nq}^{2}}{\epsilon_{0}m_{*}}$$P_{x} = {{\frac{\omega_{p}^{2}}{\left( {\omega_{0}^{2} - \omega^{2}} \right) + {j\;{\gamma\omega}}}\epsilon_{0}E\mspace{14mu} D} = {{\epsilon_{0}E} + P}}$$\epsilon = {\epsilon_{0}\left\lbrack {1 + \frac{\omega_{p}^{2}}{\left( {\omega_{0}^{2} - \omega^{2}} \right) + {j\;{\gamma\omega}}}} \right\rbrack}$ϵ = ϵ_(r) − j ϵ_(i)

Most viruses are composed of lipids, proteins, and genomes. Here aresome things known about some of the viruses:

Most lipids, V_(L)=1520 m/s

Most proteins, V_(L)=1800 m/s

Most genomes, V_(L)=1700 m/s

Due to the fact that most viruses have highly compressed genomes andtheir capsid proteins have strong tension, the effective V_(L) of atotal virus should thus be on the order of 1500-2500 m/s if the virusdoes not have a soft envelope. Using the information above, thefrequency of the [SPH, l=1, n=0] mode for a 100 nm spherical virus to betens of GHz can be estimated.

Higher order modes (i.e. SPH, l=1, n=1) may also be observed in themicrowave resonance absorption spectra at twice the frequency of dipole[SPH, l=1, n=0] mode, but at lower amplitude because dipole mode is astronger coupling.

The torsional modes which are orthogonal to the spheroidal modes arepurely transverse in nature and independent of the material property.

The torsional modes are defined for

≥1

The spheroidal modes are characterized by

≥0

=0 is the symmetric breathing mode (purely radial and produces polarizedspectra)

=1 is the dipolar mode

=2 is the quadrupole mode (produces partially depolarized spectra)

Spheroidal modes for even l (i.e.,

=0 and

=2) are Raman active

The lowest eigenfrequencies for n=0 for both spheroidal (

≥0) and torsional (

≥0) modes correspond to the surface modes

The lowest eigenfrequencies for n=1 for both spheroidal (

≥0) and torsional (

≥0) modes correspond to inner modes.

Lab Experiments

To identify the mechanical vibrations, Microwave Resonance spectralmeasurements on the viruses are performed. To do that the prepareviruses are prepared in a manner where they are cultured, isolated,purified, and then preserved in phosphate buffer saline liquids at PH of7.4 at room temperature. In each measurement, one microliter viralsolution is taken by a micropipette and uniformly dropped on a coplanarwaveguide apparatus. The guided microwaves should be incident on thevirus-containing solution. The reflection S₁₁ and transmission S₂₁parameters are recorded simultaneously using a high bandwidth networkanalyzer (one that can measure from few tens of MHz to tens of GHz). Themicrowave attenuation spectra can be evaluated by IS₁₁I²+IS₂₁I². Theattenuation spectrum of the buffer liquids is also measured with thesame volume on the same device to compare the attenuation spectra of thebuffer solutions with and without viruses, and deduce the microwaveattenuation spectra of the viruses and identify the dominant resonanceand spurious resonances of the specific virus.

Charge Distribution

There are electrostatic interactions in the virus due to the ionicatmospheres surrounding the viruses. Virus architecture, cellattachment, and other features are dependent on interactions between andwithin viruses and other structural components.

The charges of the amino acids on the surface of the capsomeres areknown. However, to describe any charge distribution, the spatial regionin which such a distribution resides must be identified and thenquantify its geometry via multipolar moments. This minimal set ofparameters includes the average size and thickness of the capsid, thesurface charge density, and surface dipole density magnitude of thecharge distribution.

There are two simple models most widely used: a single, infinitely thincharged shell of radius R_(M) and surface charge density σ, and two thinshells of inner and outer radius R_(in) and R_(out) (giving a capsidthickness of δ_(M)=R_(out)−R_(in)), carrying surface charges of σ_(in)and σ_(out). These are referred to as the single-shell and double-shellmodel, respectively. Besides the monopole (total) charge distribution,the dipole distribution on such model capsids can also be considered.Gaussian surfaces within and outside of the virus may be used tocalculate the electric field inside and outside of the virus.

Electrostatic interactions are responsible for the assembly of sphericalviral capsids. The charges on the protein subunits that make the viralcapsid mutually interact and induce electrostatic repulsion actingagainst the assembly of capsids. Thus, attractive protein-proteininteractions of non-electrostatic origin must act to enable the capsidformation. The interplay of repulsive electrostatic and attractiveinteractions between the protein subunits result in the formation ofspherical viral capsids of a specific radius. Therefore, the attractiveinteractions must depend on the angle between the neighboring proteinsubunits (i.e. on the mean curvature of the viral capsid) so thatspecific angles are preferred imposed.

Viruses can be useful, and it is possible that there are no forms oflife immune to the effect of viruses, which may be advantageous in thefight against diseases caused by microorganisms susceptible to viruses,such as bacteria. There are even viruses that initiate their “lifecycle”exclusively in combination with some other viruses and bacteria, often“stealing” the protein material of those viruses and diverting thecellular processes they initiated to their own advantage. The virusesare thus parasites even of their own kind. Although the exact nature(the shape and the genome) of only about a hundred viruses are known, itseems that they are almost as diverse as life itself. The viruses arethen only truncated, indexed, crippled representation of life and theycan barely be classified as life. To a physicists or virologists, theyare hetero-macromolecular complexes (complexes of viral proteins and thegenome molecule DNA or RNA) that are reasonably stable in extra-cellularconditions and that initiate a complicated sequence of molecularinteractions and transformations once they enter a cell. Therefore, avirus must somehow “encode” the crucial steps of its replication processbased on its structure. For example, the proteins that make itsprotective shell (virus capsid) must have such geometric/chemicalcharacteristics as to activate the appropriate receptors on the cellmembrane so that they can attach to and penetrate its interior. Thevirus needs to be sufficiently stable in the extracellular conditions,yet sufficiently unstable once it enters the cell, so that it candisassemble and deliver its genome molecule to the cellular replicationmachinery. Once it fulfills walking this tightrope of emerginginstability, the manufacturing of virus components in the cell proceeds,leading eventually to new viruses. In this view, virus has a structurein terms of the electrostatic and statistical mechanics of interactions.The major contribution to the energy of protein-genome packaging is ofelectrostatic nature.

As described above with respect to FIG. 3 , all viruses are made of twoessential parts: protein coating or a capsid and viral genome (of DNA orRNA type) situated in the capsid interior. There are also viruses thatin addition to these two essential components need an additional“wrapper”, i.e. a piece of cellular membrane, to function properly andfuse with the cellular membrane surface. These viruses are referred toas enveloped (in contrast to non-enveloped viruses which do not have amembrane coating). Because of restrictions on the length of their genomeencoding the viral shell proteins, the virus capsids are made of manycopies of one or at most a few types of proteins which are arranged in ahighly symmetrical manner.

Spherical viruses, also called icosahedral viruses, show mostlyicosahedral order and the proteins that make them can be arranged in theclusters of five (pentamers) or six (hexamers). This arrangement may beonly conceptual but may also have a physical meaning that theinteractions in clusters (capsomeres) are somewhat stronger than theinteractions between the clusters (Caspar-Klug CK classification). Crickand Watson concluded that nearly all viruses can be classified either asnearly spherical, i.e. of icosahedral symmetry, or elongated of helicalsymmetry.

Referring now to FIG. 4 , there is illustrated the geometry behind theCaspar-Klug (CK) classification of viruses. Icosahedral viruses thatobey the CK principle can be “cut out” of the lattice of proteinhexamers 402, as shown in FIG. 4 . Upon folding of the cut-out piece,twelve of protein hexamers are transformed in pentamers. The CK virusesare described with two integers, h=2 and k=1 in the case shown, whichparametrize the vector A 404. The T-number of the capsid is related to hand k as T=h²+hk+k², and the number of protein subunits is 60T. Thebottom row of images displays the CK structures with T=3, 4, 7, and 9(from left to right).

Icosahedral viruses tend to look more polyhedral when larger. There arealso non-icosahedral viruses that do not fit in the CK classification.For example, capsids of some bacteriophages (viruses that infect onlybacteria) are “elongated” (prolate) icosahedra. Capsids of some plantviruses are open and hollow cylinders and their genome molecule issituated in the empty cylindrical space formed by proteins. HIV virus isalso non-icosahedral but is not an elongated icosahedron. Its capsidlooks conical, being elongated and narrower on one side. Furthermore,even when the viruses are spherical, it is sometimes difficult toclassify them according to the CK scheme and the typicalpentamer-hexamer ordering is not obvious. Non-icosahedral capsids can beobserved in experiments as kinetically trapped structures, alsosecond-order phase transitions on spherical surfaces allowed scientiststo also classify those capsids that do not show a clear pentamer-hexamerpattern. A classification scheme based on the notion that the simplestcapsid designs are also the fittest resulted in found in numericalsimulations. Some viruses are multi-layered, i.e. they consist ofseveral protein capsids each of which may be built from differentproteins. Each of these capsid layers may individually conform to the CKprinciple. Alternatives to the CK classification have recently beenproposed that apparently contain the CK shapes as the subset of allpossible shapes. Application of the Landau theory of a “periodic table”of virus capsids that also uncovers strong evolutionary pressures. Oneof the consequences of icosahedral symmetry is that it puts restrictionson the number of proteins that can make up a spherical virus shell. Itlimits this number to 60 times the structural index T that almost alwaysassumes certain “magic” integer values T=1, 3, 4, 7, . . . . There is acertain universality in the size of capsid proteins.

By analyzing more than 80 different viruses (with T numbers from 1 to25), scientists have found that the area of a protein in a capsid isconserved and amounts to 25 nm². The thickness of the protein, i.e. thethickness of the virus capsid in question varies more but is typicallyin the interval 2 to 5 nm. The “typical” virus protein can thus beimagined as a disk/cylinder (or prism) of mean radius 3 nm andthickness/height 3 nm. In some viruses these “disks” have positivelycharged protein “tails” that protrude in the capsid interior and whoserole is to bind to a negatively charged genome molecule (typicallyssRNA). There is some universality in the distribution of charges alongand within the virus capsid.

Referring now to FIG. 5 , the calculated representation of the chargedistribution on the capsid of a virus (ssRNA): (a) the isosurface ofpositive charge (single color pigment); (b) the isosurface of negativecharge (dotted pattern) and (c) the combined isosurfaces of positive andnegative charges shown in the capsid cut in half so that its interior isseen. On the left-hand side of the image in panel (c) (the left of thewhite vertical line), the (cut) isosurface of negative charge (dottedpattern) is translated infinitesimally closer to the viewer, while it isthe opposite on the right-hand side of the image.

While Caspar-Klug dipoles corresponding to a bimodal distribution ofpositive charges on the inside and negative charges on the outside ofthe capsid can be observed, but this is certainly not a rule asmono-modal, distributions can also be observed. The in-plane angulardistribution of charges along the capsid thickness also showscomplicated variations within the constraints of the icosahedralsymmetry group. Finally, the magnitude of the charges on the surface ofthe capsomeres is regulated by the dissociation equilibrium while forthe buried charges it would have to be estimated from quantum chemicalcalculations. The virus genome molecule codes for the proteins of thecapsid, but also for other proteins needed in the process of virusreplication, depending on a virus in question. ssRNA viruses need tocode for protein that replicates the virus ssRNA and some viruses alsoencode the regulatory proteins that are required for correct assemblyand the proteins required for release of viruses from the infected cell.The amount of information that is required constrains the length of thegenome molecule.

Besides their efficient assembly mechanism, viruses also present uniquemechanical properties. During the different stages of infection, theviral capsid undergoes changes switching from highly stable states,protecting the genome, to unstable states facilitating genome release.Thus, capsids play a major role in the viral life cycle, and anunderstanding of their meta-stability and conformational plasticity iskey to deciphering the mechanisms governing the successive steps inviral infection. In addition, viral mechanical properties, such aselasticity/deformability, brittleness/hardness, material fatigue, andresistance to osmotic stress, are of interest in many areas beyondvirology; for instance, soft matter physics, (bio)nanotechnology, andnanomedicine.

In the simplest scenario of capsid formation, the functional capacity toself-assemble resides in the primary amino acid sequence of the capsidproteins (CPs) and, hence, the folded structure of the viral proteinsubunits. Thus, the assembly process is solely driven by protein-proteinand, for co-assembly with viral nucleic acids, protein-genomeinteractions. The probability of formation of a highly complex structurefrom its elements is increased, or the number of possible ways of doingit diminished, if the structure in question can be broken down in afinite series of successively smaller substrates. One of the mainchallenges of this process is that all viral proteins must encounter andassemble in the crowded environment of cells, where ˜200 mg/mL ofirrelevant, cellular, proteins are present. An additional challenge tocapsid formation is the fact that the packaging must be selective toencapsidate the viral genome, discriminating between cellular and viralgenetic material, thus ensuring infectivity. Clearly, viruses have foundstrategies to overcome these challenges, and recent literature hasreviewed different aspect of viral assembly.

Referring now to FIG. 6 , with microwave resonant absorption,electromagnetic energy at a specific microwave frequency is used at step602, which is determined by the diameters of the virus (80 to 120 nm).The diameter of cells in human body is about 50 μm which is 500× largerthan the size of the virus. Therefore, it would not have much effect onhuman cells. The diameter of each virus remains unchanged during virusmutation, which guarantees the feasibility of microwave resonantabsorption. Since a certain level of power density is needed to destroythe virus, we need to concentrate microwave power at a specific distanceusing high-gain or focusing antennas at step 604. The concentrated poweris then focused on the virus at step 606. Examples of manners forfocusing the signal include designed high-gain antenna arrays with patcharrays, such as those described below, where the radiated fields arecollimated for short distances. As for the focusing antennas, reflectorantennas and lens antennas, including the microwave counterparts of theoptical lens, are two major antenna types, and both are capable ofconcentrating microwave power onto a specific point or collimating thefields to a specific far-field direction. However, conventionalreflector antennas and lens antennas are very bulky, heavy, andexpensive to be used in a handheld device. To that end, array antennasare used that are designed to transmit Hermite-Gaussian beams to rupturethe Cartesian grid structure of the capsid proteins. Conical and hornantennas with phase or amplitude masks may also be used to achieve theproper gain as well as the proper Hermite Gaussian or Laguerre Gaussianbeams. Since the system is operating in high frequency GHz, 2×2, 4×4,8×8 or even 16×16 arrays may be used. FIGS. 7 a-7 d illustrate thedifferent modes of HG for illustration with respect to HG m=5, n=5(FIGS. 7 a, 7 c ) and HG m=6, n=4 (FIGS. 7 b, 7 d ).

Referring now to FIG. 8 , there is illustrated the optoacousticgeneration of a helicoidal ultrasonic beam is demonstrated. Such anultrasonic “doughnut” beam has a pressure amplitude minimum in thecenter along its entire longitudinal extension, and it carries orbitalangular momentum. It is produced by illuminating at step 802 a speciallystructured absorbing surface in a water tank with pulsed laser light.The absorbing surface has a profile with a screw dislocation, like thetransverse cross-sectional surface of a helix. Upon illumination withmodulated light, a correspondingly prepared absorber generates at step804 an ultrasonic wave with the desired phase discontinuity in its wavefront, which propagates through the water tank and is detected at step806 with spatial resolution using a scanning needle hydrophone. Thissituation can be viewed as the optoacoustic realization of a diffractiveacoustical element. The method can be extended to tailor otoacousticallygenerated ultrasonic waves in a customized way.

Referring now to FIG. 9 , there is illustrated various manners forgenerating a beam having orbital angular momentum applied thereto. Aseries of plane waves 902 within a beam having an intensity 904 areprocessed using one of a variety of techniques 906 in order to generatethe OAM infused beam 908 including an altered intensity profile 910. Thevariety of techniques 906 can include a spiral phase plate 912 thatchanges the phase, but for radio frequencies, a phase hologram 914 or aamplitude hologram 916 may be applied to photonic signals. Each of thesetechniques have been more fully described hereinabove. Referring nowalso to FIG. 10 , using the techniques for applying the orbital angularmomentum to create the OAM beam 908 as described in FIG. 9 , the OAMbeam 908 may be focused on to a Covid-19 virus 1002 to induce resonancestherein as described herein.

FIG. 11 illustrates the manner in which various beams having differentOAM values may be applied to a Covid-19 virus 1102. In a firstembodiment, plane waves 1104 having no OAM value applied thereto arefocused on the virus 1102. As no OAM value is applied via the planewaves 1104 no resonance would be generated within the virus 1102. Inorder to apply a resonance to the virus 1102, differing values of OAMmay be applied to a beam that is focused on the virus 1102. Beam 1106has an OAM value of I=1 focused on the virus 1102 while beam 1108applies an OAM value of I=2 to the virus, and beam 1110 applies and OAMvalue of I=3 to the virus 1102. Each of the OAM values will have adifferent effect on the generation of a resonance within the virus 1102in order to inactivate the virus, and particular OAM values may provemore or less effective depending upon the situation.

FIG. 12 illustrates the application of different Hermite Gaussian modesto a virus 1202 such as those described previously with respect to FIG.7 . A first Hermite Gaussian beam 1204 having characteristics m=5, n=5and a second Hermite Gaussian beam 1206 having characteristics m=6, n=4may be applied to viruses 1202 in order to induce resonances therein toinactivate or destroy the viruses.

As discussed previously, one manner for applying the generated beams toa virus is by the use of patch antennas is generally illustrated in FIG.13 . FIG. 14 illustrates the manner in which various patch antennas ofdifferent sizes may be used to provide different OAM values forapplication to a virus. FIG. 14 illustrates the provision of a +1, −1,+2 and −2 OAM values from different sized patch antennas. FIG. 15illustrates the different intensity signatures provided at distances of5 cm, 12.5 cm and 25 cm. Patch antennas can be used for generating bothHermite Gaussian, Laguerre Gaussian and other types of beams in order toinduce resonances with in viruses to which the beams are applied.

Referring now to FIG. 16 , there is illustrated a general block diagramof the circuitry for generating an OAM beam 1602 for application to avirus such as Covid-19. Plane wave signal generator 1604 generates aplane wave signal 1606 that has no OAM values applied thereto. Thesignal may be optical or RF in nature depending upon the particularapplication. The plane wave signal 1606 is an applied to an OAM signalgenerator 1608 along with an OAM control signal 1610. Responsive to theOAM control signal 1610 and the plane wave signal 1606, the OAM signalgenerator 1608 generates an OAM beam 1612 in accordance with the OAMvalue or values indicated by the OAM control signal 1610. The controlsignals are established based on the OAM values needed to induce aresonance within a virus that will destroy the virus. The OAM beam 1612is then provided to focusing circuitry 1614 which may be used forfocusing a beam 1602 onto a particular location. The focusing circuitry1614 may comprise components such as patch antennas, patch antennaarrays, conical antennas, or antennas or any of the other means forapplying OAM beams that are described hereinabove.

FIG. 17 illustrates one embodiment wherein the above circuitry isimplemented within a cell phone handset 1702. In this embodiment, thecell phone includes a camera 1704, a flashlight 1706 and thesterilization beam 1708. The sterilization beam 1708 is generated in themanner discussed above and the cell phone handset 1702 may bemanipulated to locate the beam on various surfaces, items and areas 1710in order to sterilize them.

In an alternative embodiment illustrated in FIG. 18 , a handheldflashlight 1702 includes the above described circuitry. The flashlight1802 generates the sterilization beam 1804 which may then be playedacross surfaces, items and areas 1806 in order to sterilize them usingthe resonance techniques described herein.

FIG. 19 illustrates a further embodiment wherein the circuitry of FIG.16 is implemented within a handheld lightbar 1902. An individual mayhold the lightbar and direct the sanitizing beam 1904 onto varioussurfaces 1906 in order to sanitize them from the Covid-19 virus throughinduced resonance by the sanitizing beam.

Alternatively, the circuitry of FIG. 16 may be implemented within aflorescent ceiling light 2002 as shown in FIG. 20 . Florescent ceilinglight 2002 would generate the sanitizing beam 2004 that would be shinedover the entire room in which the light fixtures were installed. Thiswould enable the inducement of resonances within any viruses locatedwithin the room. The fluorescent light fixture 2002 could be mounted ona ceiling 2006 as illustrated in FIG. 20 or on a wall 2102 installed asa florescent light 2002 as illustrated in FIG. 21 .

Finally, as illustrated in FIG. 22 , the circuitry of FIG. 16 could beimplemented with an incandescent light bulb 2202. The incandescent bulb2202 would generate a sanitizing beam 10904 that would sanitize anentire room in which the incandescent bulb or bulbs were installed.

New OAM and Matter Interactions with Graphene Honeycomb Lattice

Due to crystalline structure, graphene behaves like a semi-metallicmaterial, and its low-energy excitations behave as massless Diracfermions. Because of this, graphene shows unusual transport properties,like an anomalous quantum hall effect and Klein tunneling. Its opticalproperties are strange: despite being one-atom thick, graphene absorbs asignificant amount of white light, and its transparency is governed bythe fine structure constant, usually associated with quantumelectrodynamics rather than condensed matter physics.

Current efforts in study of structured light are directed, on the onehand, to the understanding and generation of twisted light beams, and,on the other hand, to the study of interaction with particles, atoms andmolecules, and Bose-Einstein condensates.

Thus, as shown in FIG. 23 , an OAM light beam 2302 may be interactedwith graphene 2304 to enable the OAM processed photons to alter thestate of the particles in the graphene 2304. The interaction of graphene2304 with light 2302 has been studied theoretically with differentapproaches, for instance by the calculation of optical conductivity, orcontrol of photocurrents. The study of the interaction of graphene 2304with light carrying OAM 2302 is interesting because the world is movingtowards using properties of graphene for many diverse applications.Since the twisted light 2302 has orbital angular momentum, one mayexpect a transfer of OAM from the photons to the electrons in graphene2304. However, the analysis is complicated by the fact that thelow-lying excitations of graphene 2304 are Dirac fermions, whose OAM isnot well-defined. Nevertheless, there is another angular momentum, knownas pseudospin, associated with the honeycomb lattice of graphene 2304,and the total angular momentum (orbital plus pseudospin) is conserved.This will work in a similar manner when OAM beams interact with viruses.

The interaction Hamiltonian between OAM and matter described herein canbe used to study the interaction of graphene with twisted light andcalculate relevant physical observables such as the photo-inducedelectric currents and the transfer of angular momentum from light toelectron particles of a material such as a virus.

The low-energy states of graphene are two-component spinors. Thesespinors are not the spin states of the electron, but they are related tothe physical lattice. Each component is associated with the relativeamplitude of the Bloch function in each sub-lattice of the honeycomblattice. They have a SU(2) algebra. This degree of freedom ispseudospin. It plays a role in the Hamiltonian like the one played bythe regular spin in the Dirac Hamiltonian. It has the same SU(2) algebrabut, unlike the isospin symmetry that connects protons and neutrons,pseudospin is an angular momentum. This pseudospin would be pointing upin z (outside the plane containing the graphene disk) in a state whereall the electrons would be found in A site, while it would be pointingdown in z if the electrons were located in the B sub-lattice.

Interaction Hamiltonian OAM with Graphene lattice (Honeycomb)

Referring now to FIG. 24 , there is illustrated a Graphene lattice in aHoneycomb structure. If T₁ and T₂ are the primitive vectors of theBravais lattice and k and k′ are the corners of the first Brillouinzone, then the Hamiltonian is:

${H_{0}(k)} = {t\begin{bmatrix}0 & {1 + e^{{- {ik}} \cdot T_{2}} + e^{{- {ik}} \cdot {({T_{2} - T_{1}})}}} \\{1 + e^{{ik} \cdot T_{2}} + e^{{ik} \cdot {({T_{2} - T_{1}})}}} & 0\end{bmatrix}}$The carbon atom separation on the lattice is a=1.42 Å. If this matrix isdiagonalized, the energy Eigen values representing energy bands ofgraphene is obtained.

${E_{\pm}(k)} = {{\pm t}\sqrt{2 + {2\mspace{14mu}{\cos\left( {\sqrt{3}k_{y}a} \right)}} + {4\mspace{14mu}{\cos\left( {\frac{\sqrt{3}}{2}k_{y}a} \right)}\mspace{14mu}{\cos\left( {\frac{3}{2}k_{x}\alpha} \right)}}}}$If k=K+(q _(x) ,q _(y))

Then

${H_{0}^{K}(q)} = {\frac{3{ta}}{2}\begin{pmatrix}0 & {q_{x} + {iq}_{y}} \\{q_{x} - {iq}_{y}} & 0\end{pmatrix}\mspace{14mu}{for}\mspace{14mu} q_{x}a\mspace{14mu}\text{<<}\mspace{14mu} 1}$q_(y)a  <<  1

So for 2D-Hamiltonian:

H₀^(α)(q) = ℏ v_(f)α(σ_(x)q_(x) − σ_(y)q_(y))  σ = (σ_(x), σ_(y))  Pauli  matricesα = ±1

Where Fermi velocity equals:

$v_{f} = {\frac{3{at}}{2\hslash} \sim {300\mspace{14mu}{times}\mspace{14mu}{slower}\mspace{14mu}{than}\mspace{14mu} c}}$The Eigen states of these Hamiltonians are spinors with 2-componentswhich are 2 elements of lattice base.

For circular graphene of radius r₀, the low-energy states can be foundin cylindrical coordinates with

${\Psi_{mv}\left( {r,\theta} \right)} = {{\frac{N_{mv}}{2\pi}{J_{m}\left( {q_{mv}r} \right)}e^{{im}\;\theta}\mspace{14mu} q_{mv}} = \frac{x_{mv}}{r_{0}}}$where x_(mv) has a zero of J_(m)(X)

To study the interaction of graphene with OAM beam, the z-component ofOAM operator

$L_{z} = {{- i}\;\hslash\frac{d}{d\;\theta}1}$is examined and then the commutation relationship between Hamiltonianand OAM is:[H ₀ ^(α) ,L _(z) ]−iℏv _(f)α(σ_(x) P _(x)+σ_(y) P _(y))

To construct a conserved angular momentum, pseudospin is added to L_(z)and the total angular momentum is:

$J_{z}^{\alpha} = {L_{z} - {\alpha\frac{\hslash}{2}\sigma_{z}}}$

This operator does commute with Hamiltonian andJ _(z)Ψ_(mv,k)=(m+½)ℏΨ_(mv,k) near k or k′Interaction Hamiltonian

We know the vector potential for OAM beam in Coulomb gauge is:

${A\left( {r,t} \right)} = {{A_{0}{e^{i{({{q_{z}z} - {\omega\; t}})}}\left\lbrack {{\epsilon_{\sigma}{J_{l}({qr})}e^{{il}\;\theta}} - {\sigma\; i\hat{z}\frac{q}{q_{z}}{J_{l + \sigma}\left( q_{r} \right)}e^{{i{({l + \sigma})}}\theta}}} \right\rbrack}} + \cdots}$where ∈_(σ)={circumflex over (x)}+iσŷ polarization vectors σ=±1The radial part of the beam are Bessel functions J_(l)(qr) andJ_(l+r)(qr). A Laguerre-Gaussian function could also be used instead ofBessel functions. Here q_(z) and q are for structured light and q_(x),q_(y), q_(mv) are for electrons.

To construct the interaction Hamiltonian:{right arrow over (P)}→{right arrow over (P)}+e{right arrow over (A)}Then

H^(α) = ℏ v_(f)α(σ_(x)q_(x) − σ_(y)q_(y)) + ev_(f)α(σ_(x)A_(x) − σ_(y)A_(y)) = H₀^(α) + H_(int)^(α)Since Graphene is 2D, there are only x, y values of the EM field.

At z=0 (Graphene disk), the vector potential is:A(r,θ,t)=A ₀({circumflex over (x)}+iσŷ)e ^(−iωt) J _(l)(qr)e ^(ilθ)+ . ..

Let's have A₊=A₀e^(−iωt)J_(l)(qr)e^(ilθ) Absorption of 1 photonA⁻=A₀e^(−iωt)J_(l)(qr)e^(−ilθ) emission of 1 photon

Then the interaction Hamiltonian close to a Dirac point a is:

$H_{int}^{\alpha,\sigma} = {{ev}_{f}\begin{bmatrix}0 & {{\left( {\sigma - \alpha} \right)A_{+}} + {\left( {\alpha + \sigma} \right)A_{-}}} \\{{\left( {\alpha + \sigma} \right)A_{+}} + {\left( {\alpha - \sigma} \right)A_{-}}} & 0\end{bmatrix}}$For α=1 (near K) and σ=+1

$H_{int}^{K +} = {2{{ev}_{f}\begin{bmatrix}0 & A_{-} \\A_{+} & 0\end{bmatrix}}}$For α=−1 (near K′) and σ=+1

$H_{int}^{K^{\prime} +} = {2{{ev}_{f}\begin{bmatrix}0 & A_{+} \\A_{-} & 0\end{bmatrix}}}$Then the transition matrix becomes:

M_(if) = ⟨iH_(int)f⟩ = ⟨c, m^(′), v^(′), αH_(int)^(ασ)v, m, v, α⟩ = ∫Ψ_(m^(′)v^(′)α)^(c^(†))(r, θ)H_(int)^(ασ)(r, θ)Ψ_(mv α)^(v)(r, θ)rdrd θWhere near K:

$\Omega_{mvK}^{v} = {{\frac{1}{\sqrt{2}}\begin{pmatrix}\Psi_{{m + 1},v} \\{i\;\Psi_{mv}}\end{pmatrix}\mspace{14mu}\Psi_{mvK}^{c}} = {\frac{1}{\sqrt{2}}\begin{pmatrix}\Psi_{{m + 1},v} \\{{- i}\;\Psi_{mv}}\end{pmatrix}}}$

Where

${\Psi_{mv}\left( {r,\theta} \right)} = {\frac{N_{mv}}{2\pi}{J_{m}\left( {q_{mv}r} \right)}e^{{im}\;\theta}}$as before

$W_{if} = {\frac{2\pi}{\hslash}{M_{if}}^{2}{\rho(E)}\mspace{14mu}{transition}\mspace{14mu}{rate}}$Ultraviolet Sterilizing

The above described techniques can also be combined with ultravioletsterilization techniques to improve the virus destruction capabilities.Ultraviolet light is the low wavelength part of electromagneticspectrum. X-rays and gamma rays are even shorter wavelength and visiblelight and radio are higher wavelength than ultraviolet. However, withinultraviolet, we have varying wavelengths decreasing from UVA, UVB, UVCand UVV.

UVC light has wavelengths between 200 to 280 nm and has been usedextensively for more than 40 years in disinfecting drinking water,wastewater, air, pharmaceutical products, and surfaces against a wholesuite of human pathogens. All bacteria and viruses tested to date (manyhundreds over the years, including other coronaviruses) respond to UVdisinfection. Some organisms are more susceptible to UVC disinfectionthan others, but all tested so far do respond at the appropriate doses.

COVID-19 infections can be caused by contact with contaminated surfacesand then touching facial areas. Minimizing this risk is key becauseCOVID-19 virus can live on plastic and steel surfaces for up to 3 days.Normal cleaning and disinfection may leave behind some residualcontamination, which UVC can treat suggesting that a multipledisinfectant approach is sensible.

In the cases where the UVC light cannot reach a particular pathogen,that pathogen will not be disinfected. However, in general, reducing thetotal number of pathogens reduces the risk of transmission. The totalpathogenic load can be reduced substantially by applying UV to the manysurfaces that are readily exposed, as a secondary barrier to cleaning,especially in hurried conditions. This would be a relativelystraight-forward matter of illuminating the relevant surfaces with UVClight, for example the air and surfaces around/in rooms and personalprotective equipment.

UVC light inactivates or kills at least two other coronaviruses that arenear-relatives of the COVID-19 virus: 1) SARS-CoV-1 and 2) MERS-CoV. Animportant caveat is this inactivation has been demonstrated undercontrolled conditions in the laboratory. The effectiveness of UV lightin practice depends on factors such the exposure time and the ability ofthe UV light to reach the viruses in water, air, and in the folds andcrevices of materials and surfaces.

Like any disinfection system, UVC devices must be used properly to besafe. This UVC light is much “stronger” than normal sunlight and cancause a severe sunburn-like reaction to skin and could damage the retinaof eye, if exposed. Some devices also produce ozone as part of theircycle, others produce light and heat like an arc welder, others moveduring their cycles. Hence, general machine-human safety needs to beconsidered with all disinfection devices, and these considerationsshould be addressed in the operations manual, in the user training, andappropriate safety compliance.

Electromagnetic (EM) waves carry spin angular momentum (SAM). Its analogin classical electrodynamics is polarization (linear or circular).However, a new property of photons was recently discovered that relateto electromagnetic (EM) vortices. Such a vortex beam has specifichelicity and an associated angular momentum which is “orbital” innature. This orbital angular momentum (OAM) of the beam is called a“twisted” or “helical” property of the beam. Current spectroscopytechniques involve circularly polarized light in which a plane polarizedstate is understood as a superposition of circular polarizations withopposite handedness. The right- and left-handedness of circularlypolarized light indicates its SAM. However as shown in FIG. 25 , ingeneral an EM beam can be engineered to have both SAM 2502 and OAM 2504and such beams are called vector beams 2506. The vector beams 2506 canbe used for new spectroscopy techniques with specific interactionsignatures with matter, but also for destroying, or diminishing theviability of viruses. Experiments support the existence of measurableOAM light-matter interactions using such vector beams 2506. By applyingdirected energy sources (using, for example, patch antenna arrays orphotonic sources) a virus may be killed. A miniaturized device may beused to either kill or limit the capability of the virus to infectothers. Specifically, the virus can be killed by exploiting a naturalsusceptibility of the virus to certain resonant frequencies as well as aspecific helicity of the beam corresponding to virus structure thatexhibit susceptibility.

Current airborne virus epidemic prevention efforts used in public space,includes strong chemicals, UV radiation, ultrasonic signals andmicrowaves. Viruses may be inactivated or destroyed in a number offashions. Viruses, however, can also be inactivated by generating thecorresponding resonance ultrasound vibrations of viruses (in the GHz).Therefore, there are vibrational modes of viruses in this frequencyrange. It is also known that dipolar mode of the confined acousticvibrations inside viruses can be resonantly excited by microwaves of thesame frequency with a resonant microwave absorption effect. Therefore,as shown in FIG. 26 , a structure-resonant energy transfer effect 2602from electromagnetic waves 2604 to vibrations of viruses 2606 can enabledestruction of the viruses. This means that energy transfer process 2602is an efficient way to excite the vibrational mode of the whole virusstructure due to an energy conversion of a photon into a phonon of thesame frequency. This would also include a transfer of the angularmomentum using structured vector beams that carry OAM as shown in FIG.27 .

As more fully described herein below, a model of an interactionHamiltonian which can be used to study light-matter interaction andexplore the relation between the induced stress and the field magnitudeof the microwave. Since the viruses could be inactivated when theinduced stress fractures the structure of viruses, the SRET efficiencyfrom microwaves to COVID-19 may be explored through measuring the virusinactivation threshold. Based on the model, the inactivation ratio ofthe virus at dipolar-mode-resonance and off-resonance microwavefrequencies as well as with different microwave powers may bedetermined. The resonant frequency, the microwave power densitythreshold for COVID-19 inactivation must be below the IEEE safetystandard to enable use of the device. The main inactivation mechanism isthrough physically fracturing the viruses, while the RNA genome is notdegraded by the microwave illumination, supporting the fact that thisapproach is fundamentally different from the microwave thermal heatingeffect. The COVID-19 virus is a 100 nm virus with different types ofproteins (spike, envelope, membrane, nucleocapsid) as well as RNA.

Referring now more particularly to FIG. 28 , there is illustrated afunctional block diagram of a system for generating the orbital angularmomentum “twist” that may be imparted to a virus. A plane wave signal2802 is provided to the transmission processing circuitry 2800. Theplane wave signal is provided to the orbital angular momentum (OAM)signal processing block 2806. The plane wave signal is provided adifferent orbital angular momentum by the orbital angular momentumelectromagnetic block 2806 depending on the virus that is to bedeactivated or destroyed. Each of the signals having an associatedorbital angular momentum are provided to an optical transmitter 2808that transmits each of the data streams having a unique orbital angularmomentum on a same wavelength.

FIG. 29 illustrates in a manner in which a single wavelength orfrequency, having two quanti-spin polarizations may provide an infinitenumber of twists having various orbital angular momentums associatedtherewith. The l axis represents the various quantized orbital angularmomentum states which may be applied to a particular signal at aselected frequency or wavelength. The symbol omega (ω) represents thevarious frequencies to which the signals of differing orbital angularmomentum may be applied. The top grid 2902 represents the potentiallyavailable signals for a left-handed signal polarization, while thebottom grid 2904 is for potentially available signals having righthanded polarization.

By applying different orbital angular momentum states to a signal at aparticular frequency or wavelength, a potentially infinite number ofstates may be provided at the frequency or wavelength. Thus, the stateat the frequency Δω or wavelength 2906 in both the left-handedpolarization plane 2902 and the right-handed polarization plane 2904 canprovide an infinite number of signals at different orbital angularmomentum states ΔI. Blocks 2908 and 2910 represent a particular signalhaving an orbital angular momentum ΔI at a frequency Δω or wavelength inboth the right-handed polarization plane 2904 and left-handedpolarization plane 2910, respectively. By changing to a differentorbital angular momentum within the same frequency Δω or wavelength2906, different signals may also be transmitted. Each angular momentumstate corresponds to a different determined current level fortransmission from the optical transmitter. By estimating the equivalentcurrent for generating a particular orbital angular momentum within theoptical domain and applying this current for transmission of thesignals, the transmission of the signal may be achieved at a desiredorbital angular momentum state.

Thus, the illustration of FIG. 29 , illustrates two possible angularmomentums, the spin angular momentum, and the orbital angular momentum.The spin version is manifested within the polarizations of macroscopicelectromagnetism and has only left and right-hand polarizations due toup and down spin directions. However, the orbital angular momentumindicates an infinite number of states that are quantized. The paths aremore than two and can theoretically be infinite through the quantizedorbital angular momentum levels.

It is well-known that the concept of linear momentum is usuallyassociated with objects moving in a straight line. The object could alsocarry angular momentum if it has a rotational motion, such as spinning(i.e., spin angular momentum (SAM) 3002), or orbiting around an axis3006 (i.e., OAM 3004), as shown in FIGS. 30A and 30B, respectively. Alight beam may also have rotational motion as it propagates. In paraxialapproximation, a light beam carries SAM 3002 if the electrical fieldrotates along the beam axis 3006 (i.e., circularly polarized light3005), and carries OAM 3004 if the wave vector spirals around the beamaxis 3006, leading to a helical phase front 3008, as shown in FIGS. 30Cand 30D. In its analytical expression, this helical phase front 3008 isusually related to a phase term of exp(i

θ) in the transverse plane, where θ refers to the angular coordinate,and

is an integer indicating the number of intertwined helices (i.e., thenumber of 2π phase shifts along the circle around the beam axis).

could be a positive, negative integer or zero, corresponding toclockwise, counterclockwise phase helices or a Gaussian beam with nohelix, respectively.

Two important concepts relating to OAM include:

1) OAM and polarization: As mentioned above, an OAM beam is manifestedas a beam with a helical phase front and therefore a twistingwavevector, while polarization states can only be connected to SAM 3002.A light beam carries SAM 3002 of ±h/2π (h is Plank's constant) perphoton if it is left or right circularly polarized and carries no SAM3002 if it is linearly polarized. Although the SAM 3002 and OAM 3004 oflight can be coupled to each other under certain scenarios, they can beclearly distinguished for a paraxial light beam. Therefore, with theparaxial assumption, OAM 3004 and polarization can be considered as twoindependent properties of light.2) OAM beam and Laguerre-Gaussian (LG) beam: In general, an OAM-carryingbeam could refer to any helically phased light beam, irrespective of itsradial distribution (although sometimes OAM could also be carried by anon-helically phased beam). LG beam is a special subset among allOAM-carrying beams, due to that the analytical expression of LG beamsare eigen-solutions of paraxial form of the wave equation in cylindricalcoordinates. For an LG beam, both azimuthal and radial wavefrontdistributions are well defined, and are indicated by two index numbers,

and p, of which

has the same meaning as that of a general OAM beam, and p refers to theradial nodes in the intensity distribution. Mathematical expressions ofLG beams form an orthogonal and complete basis in the spatial domain. Incontrast, a general OAM beam actually comprises a group of LG beams(each with the same

index but a different p index) due to the absence of radial definition.The term of “OAM beam” refers to all helically phased beams and is usedto distinguish from LG beams.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the radiationtransmitted by the signals. The Maxwell-Heaviside equations can berepresented as:

${\nabla{\cdot E}} = \frac{\rho}{ɛ_{0}}$${\nabla{\times E}} = {- \frac{\partial B}{\partial t}}$ ∇⋅B = 0${\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}$where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, one can derive 23symmetries/conserved quantities from Maxwell's original equations.However, there are only ten well-known conserved quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$H = {{\sum\limits_{i}{m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int{d^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}\mspace{14mu}{Hamiltonian}\mspace{14mu}\left( {{total}\mspace{14mu}{energy}} \right)}}}}$$\mspace{76mu}{{\frac{{dU}^{mech}}{dt} + \frac{{dU}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{{\hat{n}}^{\prime} \cdot S}}}} = {0\mspace{14mu}{conservation}\mspace{14mu}{of}\mspace{14mu}{energy}}}$

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

$\mspace{76mu}{p = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int{d^{3}{x\left( {E \times B} \right)}\mspace{14mu}{linear}\mspace{14mu}{momentum}}}}}}$${\frac{{dp}^{mech}}{dt} + \frac{{dp}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{n^{\hat{\prime}} \cdot T}}}} = {0\mspace{14mu}{conservation}\mspace{14mu}{of}\mspace{14mu}{linear}\mspace{14mu}{momentum}}$

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\Sigma_{i}\left( {x_{i} - x_{0}} \right)}m_{i}\gamma_{i}c^{2}} + {\frac{ɛ_{0}}{2H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}$Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

${\frac{{dJ}^{mech}}{dt} + \frac{{dJ}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{n^{\hat{\prime}} \cdot M}}}} = {0\mspace{14mu}{conservation}\mspace{14mu}{of}\mspace{14mu}{angular}\mspace{14mu}{momentum}}$

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:

J^(em) = ɛ₀∫_(V^(′))d³x^(′)(E × A) + ɛ₀∫_(V^(′))d³x^(′)E_(i)[(x^(′) − x₀) × ∇]A_(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- i}\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) its classical manifestation is wavehelicity. In general, both EM linear momentum P^(em), and EM angularmomentum J^(em)=L^(em)+S^(em) are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

${{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},$continuity equationwhere S is the Poynting vectorS=¼(E×H*+E*×H),and U is the energy densityU=¼(ε|E| ²+μ₀ |H| ²),with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{\varepsilon{❘E❘}^{2}} + {\mu_{0}{❘H❘}^{2}}} \right)}}}$

The use of the OAM of light for the metrology of glucose, amyloid betaand other chiral materials has been demonstrated using theabove-described configurations. OAM beams are observed to exhibit uniquetopological evolution upon interacting with chiral solutions within 3 cmoptical path links. It should be realized that unique topologicalevolution may also be provided from non-chiral materials. Chiralsolution, such as Amyloid-beta, glucose and others, have been observedto cause orbital angular momentum (OAM) beams to exhibit uniquetopological evolution when interacting therewith. OAM is not typicallycarried by naturally scattered photons which make use of the twistedbeams more accurate when identifying the helicities of chiral moleculesbecause OAM does not have ambient light scattering (noise) in itsdetection. Thus, the unique OAM signatures imparted by a material is notinterfered with by ambient light scattering (noise) that does not carryOAM in naturally scattered photons making detection much more accurate.Given these unique topological features one can detect the amyloid-betapresence and concentration within a given sample based upon a specificsignature in both amplitude and phase measurements. Molecular chiralitysignifies a structural handedness associated with variance under spatialinversion or a combination of inversion and rotation, equivalent to theusual criteria of a lack of any proper axes of rotation. Something ischiral when something cannot be made identical to its reflection. Chiralmolecules that are not superimposable on their mirror image are known asEnantiomers. Traditionally, engages circularly polarized light, even inthe case of optical rotation, interpretation of the phenomenon commonlyrequires the plane polarized state to be understood as a superpositionof circular polarizations with opposite handedness. For circularlypolarized light, the left and right forms designate the sign ofintrinsic spin angular momentum, ±h and also the helicity of the locusdescribed by the associated electromagnetic field vectors. For thisreason its interactions with matter are enantiomerically specific.

The continuous symmetry measure (CSM) is used to evaluate the degree ofsymmetry of a molecule, or the chirality. This value ranges from 0 to100. The higher the symmetry value of a molecule the more symmetrydistorted the molecule and the more chiral the molecule. The measurementis based on the minimal distance between the chiral molecule and thenearest achiral molecule.

The continuous symmetry measure may be achieved according to theequation:

${S(G)} = {100 \times \min\frac{1}{{Nd}^{2}}{\sum\limits_{k = 1}^{N}{❘{Q_{k} - {\hat{Q}}_{k}}❘}^{2}}}$Q_(k): The original structure{circumflex over (Q)}_(k): The symmetry-operated structureN: Number of verticesd: Size normalization factor*The scale is 0-1 (0-100):The larger S(G) is, the higher is the deviation from G-symmetry

SG as a continuous chirality measure may be determined according to:

${S(G)} = {100 \times \min\frac{1}{{Nd}^{2}}{\sum\limits_{k = 1}^{N}{❘{Q_{k} - {\hat{Q}}_{k}}❘}^{2}}}$G: The achiral symmetry point group which minimizes S(G)Achiral molecule: S(G)=0

An achiral molecule has a value of S(G)=0. The more chiral a molecule isthe higher the value of S(G).

The considerable interest in orbital angular momentum has been enhancedthrough realization of the possibility to engineer optical vortices.Here, helicity is present in the wave-front surface of theelectromagnetic fields and the associated angular momentum is termed“orbital”. The radiation itself is commonly referred to as a ‘twisted’or ‘helical’ beam. Mostly, optical vortices have been studied only intheir interactions with achiral matter—the only apparent exception issome recent work on liquid crystals. It is timely and of interest toassess what new features, if any, can be expected if such beams are usedto interrogate any system whose optical response is associated withenantiomerically specific molecules.

First the criteria for manifestations of chirality in opticalinteractions are constructed in generalized form. For simplicity,materials with a unique enantiomeric specificity are assumed—signifyinga chirality that is intrinsic and common to all molecular components (orchromophores) involved in the optical response. Results for systems ofthis kind will also apply to single molecule studies. Longer rangetranslation/rotation order can also produce chirality, as for example intwisted nematic crystals, but such mesoscopic chirality cannot directlyengender enantiomerically specific interactions. The only exception iswhere optical waves probe two or more electronically distinct,dissymmetrically oriented but intrinsically achiral molecules orchromophores.

Chiroptical interactions can be distinguished by their electromagneticorigins: for molecular systems in their usual singlet electronic groundstate, they involve the spatial variation of the electric and magneticfields associated with the input of optical radiation. This variationover space can be understood to engage chirality either through itscoupling with di-symmetrically placed, neighbouring chromophore groups(Kirkwood's two-group model, of limited application) or more generallythrough the coupling of its associated electric and magnetic fields withindividual groups. As chirality signifies a local breaking of parity itpermits an interference of electric and magnetic interactions. Even inthe two-group case, the paired electric interactions of the systemcorrespond to electric and magnetic interactions of the single entitywhich the two groups comprise. Thus, for convenience, the term ‘chiralcenter’ is used in the following to denote either chromophore ormolecule.

With the advent of the laser, the Gaussian beam solution to the waveequation came into common engineering parlance, and its extension twohigher order laser modes, Hermite Gaussian for Cartesian symmetry;Laguerre Gaussian for cylindrical symmetry, etc., entered laboratoryoptics operations. Higher order Laguerre Gaussian beam modes exhibitspiral, or helical phase fronts. Thus, the propagation vector, or theeikonal of the beam, and hence the beams momentum, includes in additionto a spin angular momentum, an orbital angular momentum, i.e. a wobblearound the sea axis. This phenomenon is often referred to as vorticity.The expression for a Laguerre Gaussian beam is given in cylindricalcoordinates:

${u\left( {r,\theta,z} \right)} = {\sqrt{\frac{2{pl}}{1 + {\delta_{0,m}{{\pi\left( {m + p} \right)}!}}}}\frac{1}{w(z)}{{{\exp\left\lbrack {{j\left( {{2p} + m + 1} \right)}\left( {{\psi(z)} - \psi_{0}} \right)} \right\rbrack}\left( \frac{\sqrt{2}r}{w(z)} \right){L_{p}^{m}\left( \frac{2r^{2}}{{w(z)}^{2}} \right)}{\exp\left\lbrack {{{- {jk}}\frac{r^{2}}{2{q(z)}}} + {im\theta}} \right\rbrack}}}}$

Here, w (x) is the beam spot size, q(c) is the complex beam parametercomprising the evolution of the spherical wave front and the spot size.Integers p and m are the radial and azimuthal modes, respectively. Theexp(imθ) term describes the spiral phase fronts.

Referring now also to FIG. 31 , there is illustrated one embodiment of abeam for use with the system. A light beam 3100 consists of a stream ofphotons 3102 within the light beam 3100. Each photon has an energy±ℏ

and a linear momentum of ±ℏk which is directed along the light beam axis3104 perpendicular to the wavefront. Independent of the frequency, eachphoton 3102 within the light beam has a spin angular momentum 3106 of ±ℏaligned parallel or antiparallel to the direction of light beampropagation. Alignment of all of the photons 3102 spins gives rise to acircularly polarized light beam. In addition to the circularpolarization, the light beams also may carry an orbital angular momentum3108 which does not depend on the circular polarization and thus is notrelated to photon spin.

Lasers are widely used in optical experiments as the source ofwell-behaved light beams of a defined frequency. A laser may be used forproviding the light beam 3100. The energy flux in any light beam 3100 isgiven by the Poynting vector which may be calculated from the vectorproduct of the electric and magnetic fields within the light beam. In avacuum or any isotropic material, the Poynting vector is parallel to thewave vector and perpendicular to the wavefront of the light beam. In anormal laser light, the wavefronts 3200 are parallel as illustrated inFIG. 32 . The wave vector and linear momentum of the photons aredirected along the axis in a z direction 3202. The field distributionsof such light beams are paraxial solutions to Maxwell's wave equationbut although these simple beams are the most common, other possibilitiesexist.

For example, beams that have l intertwined helical fronts are alsosolutions of the wave equation. The structure of these complicated beamsis difficult to visualize, but their form is familiar from the l=3fusilli pasta. Most importantly, the wavefront has a Poynting vector anda wave vector that spirals around the light beam axis direction ofpropagation as illustrated in FIG. 33 at 3302.

A Poynting vector has an azimuthal component on the wave front and anon-zero resultant when integrated over the beam cross-section. The spinangular momentum of circularly polarized light may be interpreted in asimilar way. A beam with a circularly polarized planer wave front, eventhough it has no orbital angular momentum, has an azimuthal component ofthe Poynting vector proportional to the radial intensity gradient. Thisintegrates over the cross-section of the light beam to a finite value.When the beam is linearly polarized, there is no azimuthal component tothe Poynting vector and thus no spin angular momentum.

Thus, the momentum of each photon 3102 within the light beam 3100 has anazimuthal component. A detailed calculation of the momentum involves allof the electric fields and magnetic fields within the light beam,particularly those electric and magnetic fields in the direction ofpropagation of the beam. For points within the beam, the ratio betweenthe azimuthal components and the z components of the momentum is foundto be l/kr. (where l=the helicity or orbital angular momentum; k=wavenumber 2π/λ; r=the radius vector.) The linear momentum of each photon3102 within the light beam 3100 is given by ℏk, so if we take the crossproduct of the azimuthal component within a radius vector, r, we obtainan orbital momentum for a photon 3102 of lℏ. Note also that theazimuthal component of the wave vectors is l/r and independent of thewavelength.

Referring now to FIGS. 34 and 35 , there are illustrated planewavefronts and helical wavefronts. Ordinarily, laser beams with planewavefronts 3402 are characterized in terms of Hermite-Gaussian modes.These modes have a rectangular symmetry and are described by two modeindices m 3404 and n 3406. There are m nodes in the x direction and nnodes in the y direction. Together, the combined modes in the x and ydirection are labeled HGmn 3408. In contrast, as shown in FIG. 35 ,beams with helical wavefronts 3502 are best characterized in terms ofLaguerre-Gaussian modes which are described by indices I 3503, thenumber of intertwined helices 3504, and p, the number of radial nodes3506. The Laguerre-Gaussian modes are labeled LGmn 14710. For l≠0, thephase singularity on a light beam 3100 results in 0 on axis intensity.When a light beam 300 with a helical wavefront is also circularlypolarized, the angular momentum has orbital and spin components, and thetotal angular momentum of the light beam is (l±ℏ) per photon.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the electromagneticradiation transmitted by the signals. The Maxwell-Heaviside equationscan be represented as:

$\begin{matrix}{{\nabla{\cdot E}} = \frac{\rho}{\varepsilon_{0}}} \\{{\nabla{\times E}} = {- \frac{\partial B}{\partial t}}} \\{{\nabla{\cdot B}} = 0} \\{{\nabla{\times B}} = {{\varepsilon_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}}\end{matrix}$where ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, we can derive 23symmetries/conserve quantities from Maxwell's original equations.However, there are only ten well-known conserve quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$\begin{matrix}{H = {{\sum\limits_{i}{m_{i}\gamma_{i}c^{2}}} + {\frac{\varepsilon_{0}}{2}{\int{d^{3}{x\left( {{❘E❘}^{2} + {c^{2}{❘B❘}^{2}}} \right)}}}}}} \\{{\frac{{dU}^{mech}}{dt} + \frac{{dU}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot S}}}} = 0}\end{matrix}$

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

$\begin{matrix}{P = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {\varepsilon_{0}{\int{d^{3}{x\left( {E \times B} \right)}}}}}} \\{{\frac{{dp}^{mech}}{dt} + \frac{{dp}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot T}}}} = 0}\end{matrix}$

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\sum\limits_{i}{\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{\varepsilon_{0}}{2H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\left( {{❘E^{2}❘} + {c^{2}{❘B^{2}❘}}} \right)}}}}$

Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

${\frac{{dJ}^{mech}}{dt} + \frac{{dJ}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot M}}}} = 0$

For radiation beams in free space, the EM field angular momentum Jem canbe separated into two parts:

J^(em) = ε₀∫_(V^(′)) d³x^(′)(E × A) + ε₀∫_(V^(′)) d³x^(′)E_(i)[(x^(′) − x₀) × ∇]A_(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- i}\frac{\varepsilon_{0}}{2\omega}{\int_{V^{\prime}}\,{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\frac{\varepsilon_{0}}{2\omega}{\int_{V^{\prime}}\,{d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum Sem, its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum Lem its classical manifestation is wavehelicity. In general, both EM linear momentum Pem, and EM angularmomentum Jem=Lem+Sem are radiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

${\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0$where S is the Poynting vectorS=¼(E×H*+E*×H)and U is the energy densityU=¼(ε|E| ²+μ₀ |H| ²)with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{\varepsilon{❘E❘}^{2}} + {\mu_{0}{❘H❘}^{2}}} \right)}}}$

Referring now to FIGS. 36 and 37 , there are illustrated the manner inwhich a signal and an associated Poynting vector of the signal vary in aplane wave situation (FIG. 36 ) where only the spin vector is altered,and in a situation wherein the spin and orbital vectors are altered in amanner to cause the Poynting vector to spiral about the direction ofpropagation (FIG. 37 ).

In the plane wave situation, illustrated in FIG. 36 , when only the spinvector of the plane wave is altered, the transmitted signal may take onone of three configurations. When the spin vectors are in the samedirection, a linear signal is provided as illustrated generally at 3604.It should be noted that while 3604 illustrates the spin vectors beingaltered only in the x direction to provide a linear signal, the spinvectors can also be altered in the y direction to provide a linearsignal that appears similar to that illustrated at 3604 but in aperpendicular orientation to the signal illustrated at 3604. In linearpolarization such as that illustrated at 3604, the vectors for thesignal are in the same direction and have a same magnitude.

Within a circular polarization as illustrated at 3606, the signalvectors 3612 are 90 degrees to each other but have the same magnitude.This causes the signal to propagate as illustrated at 3606 and providethe circular polarization 3614 illustrated in FIG. 36 . Within anelliptical polarization 3608, the signal vectors 3616 are also 90degrees to each other but have differing magnitudes. This provides theelliptical polarizations 3618 illustrated for the signal propagation408. For the plane waves illustrated in FIG. 36 , the Poynting vector ismaintained in a constant direction for the various signal configurationsillustrated therein.

The situation in FIG. 37 illustrates when a unique orbital angularmomentum is applied to a signal or beam. When this occurs, Poyntingvector S 3710 will spiral around the general direction of propagation3712 of the signal. The Poynting vector 3710 has three axial componentsSφ, Sp and Sz which vary causing the vector to spiral about thedirection of propagation 3712 of the signal. The changing values of thevarious vectors comprising the Poynting vector 3710 may cause the spiralof the Poynting vector to be varied in order to enable signals to betransmitted on a same wavelength or frequency as will be more fullydescribed herein. Additionally, the values of the orbital angularmomentum indicated by the Poynting vector 3710 may be measured todetermine the presence of particular materials and the concentrationsassociated with particular materials being processed by a scanningmechanism.

FIGS. 38A-38C illustrate the differences in signals having a differenthelicity (i.e., orbital angular momentum applied thereto). The differinghelicities would be indicative of differing materials and concentrationof materials within a sample that a beam was being passed through. Bydetermining the particular orbital angular momentum signature associatedwith a signal, the particular material and concentration amounts of thematerial could be determined. Each of the spiralling Poynting vectorsassociated with a signal 3802, 3804 and 3806 provides a different-shapedsignal. Signal 3802 has an orbital angular momentum of +1, signal 3804has an orbital angular momentum of +3 and signal 3806 has an orbitalangular momentum of −4. Each signal has a distinct orbital angularmomentum and associated Poynting vector enabling the signal to beindicative of a particular material and concentration of material thatis associated with the detected orbital angular momentum. This allowsdeterminations of materials and concentrations of various types ofmaterials to be determined from a signal since the orbital angularmomentums are separately detectable and provide a unique indication ofthe particular material and the concentration of the particular materialthat has affected the orbital angular momentum of the signal transmittedthrough the sample material.

FIG. 39A illustrates the propagation of Poynting vectors for variousEigen modes. Each of the rings 3920 represents a different Eigen mode ortwist representing a different orbital angular momentum. Each of thedifferent orbital angular momentums is associated with particularmaterial and a particular concentration of the particular material.Detection of orbital angular momentums provides an indication of thepresence of an associated material and a concentration of the materialthat is being detected by the apparatus. Each of the rings 3920represents a different material and/or concentration of a selectedmaterial that is being monitored. Each of the Eigen modes has a Poyntingvector 3922 for generating the rings indicating different materials andmaterial concentrations.

Topological charge may be multiplexed to the frequency for either linearor circular polarization. In case of linear polarizations, topologicalcharge would be multiplexed on vertical and horizontal polarization. Incase of circular polarization, topological charge would multiplex onleft hand and right-hand circular polarizations. The topological chargeis another name for the helicity index “I” or the amount of twist or OAMapplied to the signal. The helicity index may be positive or negative.

The topological charges 1 s can be created using Spiral Phase Plates(SPPs) as shown in FIG. 39B using a proper material with specific indexof refraction and ability to machine shop or phase mask, hologramscreated of new materials. Spiral Phase plates can transform a RF planewave (l=0) to a twisted wave of a specific helicity (i.e. l=+1).

Referring now to FIG. 40 , there is illustrated a block diagram of theapparatus for providing detection of the presence of a material andconcentration measurements of various materials responsive to theorbital angular momentum detected by the apparatus in accordance withthe principles described herein above. An emitter 4002 transmits waveenergy 4004 that comprises a series of plane waves. The emitter 4002 mayprovide a series of plane waves such as those describes previously withrespect to FIG. 32 . The orbital angular momentum generation circuitry4006 generates a series of waves having an orbital angular momentumapplied to the waves 4008 in a known manner. The orbital angularmomentum generation circuitry 4006 may utilize holograms or some othertype of orbital angular momentum generation process as will be morefully described herein below. The OAM generation circuitry 4006 may begenerated by transmitting plane waves through a spatial light modulator(SLM), an amplitude mask or a phase mask. The orbital angular momentumtwisted waves 4008 are applied to a sample material 4010 under test. Thesample material 4010 contains a material, and the presence andconcentration of the material is determined via a detection apparatus inaccordance with the process described herein. The sample material 4010may be located in a container or at its naturally occurring location innature such as an individual's body.

A series of output waves 4012 from the sample material 4010 exit thesample and have a particular orbital angular momentum imparted theretoas a result of the material and the concentration of the particularmaterial under study within the sample material 4010. The output waves4012 are applied to a matching module 4014 that includes a mappingaperture for amplifying a particular orbital angular momentum generatedby the specific material under study. The matching module 4014 willamplify the orbital angular momentums associated with the particularmaterial and concentration of material that is detected by theapparatus. The amplified OAM waves 4016 are provided to a detector 4018.The detector 4018 detects OAM waves relating to the material and theconcentration of a material within the sample and provides thisinformation to a user interface 4020. The detector 4018 may utilize acamera to detect distinct topological features from the beam passingthrough the sample. The user interface 4020 interprets the informationand provides relevant material type and concentration indication to anindividual or a recording device.

Referring now to FIG. 41 , there is more particularly illustrated theemitter 4002. The emitter 4002 may emit a number of types of energywaves 4004 to the OAM generation module 4006. The emitter 4002 may emitoptical waves 4100, electromagnetic waves 4102, acoustic waves 4104 orany other type of particle waves 4106. The emitted waves 4004 are planewaves such as those illustrated in FIG. 32 having no orbital angularmomentum applied thereto and may come from a variety of types ofemission devices and have information included therein. In oneembodiment, the emission device may comprise a laser. Plane waves havewavefronts that are parallel to each other having no twist or helicityapplied thereto, and the orbital angular momentum of the wave is equalto 0. The Poynting vector within a plane wave is completely in line withthe direction of propagation of the wave.

The OAM generation module 4006 processes the incoming plane wave 4004and imparts a known orbital angular momentum onto the plane waves 4004provided from the emitter 4002. The OAM generation module 4006 generatestwisted or helical electromagnetic, optic, acoustic or other types ofparticle waves from the plane waves of the emitter 4002. A helical wave4008 is not aligned with the direction of propagation of the wave buthas a procession around direction of propagation as shown in FIG. 42 .The OAM generation module 4006 may comprise in one embodiment a fixedorbital angular momentum generator 4202 as illustrated in FIG. 42 . Thefixed orbital angular momentum generator 4202 receives the plane waves4004 from the emitter 4002 and generates an output wave 4204 having afixed orbital angular momentum applied thereto.

The fixed orbital angular momentum generator 4202 may in one embodimentcomprise a holographic image for applying the fixed orbital angularmomentum to the plane wave 4004 in order to generate the OAM twistedwave 4204. Various types of holographic images may be generated in orderto create the desired orbital angular momentum twist to an opticalsignal that is being applied to the orbital angular momentum generator4202. Various examples of these holographic images are illustrated inFIG. 43 . In one embodiment, the conversion of the plane wave signalstransmitted from the emitter 4002 by the orbital angular momentumgeneration circuitry 4006 may be achieved using holographic images.

Most commercial lasers emit an HG00 (Hermite-Gaussian) mode 4402 (FIG.44 ) with a planar wave front and a transverse intensity described by aGaussian function. Although a number of different methods have been usedto successfully transform an HG00 Hermite-Gaussian mode 4402 into aLaguerre-Gaussian mode 4404, the simplest to understand is the use of ahologram.

The cylindrical symmetric solution upl (r,φ,z) which describesLaguerre-Gaussian beams, is given by the equation:

${u_{pl}\left( {r,\phi,z} \right)} = {{\frac{C}{\left( {1 + {z^{2}/z_{R}^{2}}} \right)^{1/2}}\left\lbrack \frac{r\sqrt{2}}{w(z)} \right\rbrack}^{l}{L_{p}^{l}\left\lbrack \frac{2r^{2}}{w^{2}(z)} \right\rbrack}{\exp\left\lbrack \frac{- r^{2}}{w^{2}(z)} \right\rbrack}{\exp\left\lbrack \frac{{- {ikr}^{2}}z}{2\left( {z^{2} + z_{R}^{2}} \right.} \right\rbrack}{\exp\left( {{- {il}}\phi} \right)} \times {}{\exp\left\lbrack {{i\left( {{2p} + l + 1} \right)}\tan^{- 1}\frac{z}{z_{R}}} \right\rbrack}}$Where z_(R) is the Rayleigh range, w(z) is the radius of the beam, L_(P)is the Laguerre polynomial, C is a constant, and the beam waist is atz=0.

In its simplest form, a computer-generated hologram is produced from thecalculated interference pattern that results when the desired beamintersects the beam of a conventional laser at a small angle. Thecalculated pattern is transferred to a high-resolution holographic film.When the developed hologram is placed in the original laser beam, adiffraction pattern results. The first order of which has a desiredamplitude and phase distribution. This is one manner for implementingthe OAM generation module 4006. An example of holographic images for usewithin a OAM generation module is illustrated with respect to FIG. 43 .

There are various levels of sophistication in hologram design. Hologramsthat comprise only black and white areas with no grayscale are referredto as binary holograms. Within binary holograms, the relativeintensities of the two interfering beams play no role and thetransmission of the hologram is set to be zero for a calculated phasedifference between zero and π, or unity for a phase difference between πand 2π. A limitation of binary holograms is that very little of theincident power ends up in the first order diffracted spot, although thiscan be partly overcome by blazing the grating. When mode purity is ofparticular importance, it is also possible to create more sophisticatedholograms where the contrast of the pattern is varied as a function ofradius such that the diffracted beam has the required radial profile.

A plane wave shining through the holographic images 1502 will have apredetermined orbital angular momentum shift applied thereto afterpassing through the holographic image 1502. OAM generator 4002 is fixedin the sense that a same image is used and applied to the beam beingpassed through the holographic image. Since the holographic image 1502does not change, the same orbital angular momentum is always applied tothe beam being passed through the holographic image 1502. While FIG. 43illustrates an embodiment of a holographic image that might be utilizedwithin the orbital angular momentum generator 4002, it will be realizedthat any type of holographic image 1502 may be utilized in order toachieve the desired orbital angular momentum within an beam being shinedthrough the image 1502.

In another example of a holographic image illustrated in FIG. 45 , thereis illustrated a hologram that utilizes two separate holograms that aregridded together to produce a rich number of orbital angular momentum(l). The superimposed holograms of FIG. 45 have an orbital angularmomentum of l=1 and l=3 which are superimposed upon each other tocompose the composite vortex grid 4502. The holograms utilized may alsobe built in a manner that the two holograms are gridded together toproduce a varied number of orbital angular momentums (l) not just on aline (l=+1, l=0, l=−1) but on a square which is able to identify themany variables more easily. Thus, in the example in FIG. 45 , theorbital angular momentums along the top edge vary from +4 to +1 to −2and on the bottom edge from +2 to −1 to −4. Similarly, along the leftedge the orbital angular momentums vary from +4 to +3 to +2 and on theright edge from −2 to −3 to −4. Across the horizontal center of thehologram the orbital angular momentums provided vary from +3 to 0 to −3and along the vertical axis vary from +1 to 0 to −1. Thus, dependingupon the portion of the grid a beam may pass through, varying orbitalangular momentum may be achieved.

Referring now to FIG. 46 , in addition to a fixed orbital angularmomentum generator, the orbital angular momentum generation circuitry4006 may also comprise a tunable orbital angular momentum generatorcircuitry 4602. The tunable orbital angular momentum generator 4602receives the input plane wave 4004 but additionally receives one or moretuning parameters 4604. The tuning parameters 4604 tune the tunable OAMgenerator 4602 to apply a selected orbital angular momentum so that thetuned OAM wave 4606 that is output from the OAM generator 4602 has aselected orbital angular momentum value applied thereto. This can proveuseful in inducing different resonances in different types of viruses.

This may be achieved in any number of fashions. In one embodiment,illustrated in FIGS. 46 and 26 , the tunable orbital angular momentumgenerator 4602 may include multiple hologram images 2602 within thetunable OAM generator 4602. The tuning parameters 4604 enable selectionof one of the holographic images 2602 in order to provide the desiredOAM wave twisted output signal 4606 through a selector circuit 2604.Alternatively, the gridded holographic image such as that described inFIG. 45 may be utilized and the beam shined on a portion of the griddedimage to provide the desired OAM output. The tunable OAM generator 4602has the advantage of being controlled to apply a particular orbitalangular momentum to the output orbital angular momentum wave 4606depending upon the provided input parameter 4604. This enables thepresence and concentrations of a variety of different materials to bemonitored, or alternatively, for various different concentrations of thesame material to be monitored or for different viruses to be attacked.

Referring now to FIG. 47 , there is more particularly implemented ablock diagram of a tunable orbital angular momentum generator 4602. Thegenerator 4602 includes a plurality of holographic images 4702 forproviding orbital angular momentums of various types to a provided lightsignal. These holographic images 4702 are selected responsive to aselector circuitry 4704 that is responsive to the input tuningparameters 4604. The selected filter 4706 comprises the holographicimage that has been selected responsive to the selector controller 4704and receives the input plane waves 4004 to provide the tuned orbitalangular momentum wave output 4606. In this manner, signals having adesired orbital angular momentum may be output from the OAM generationcircuitry 4006.

Referring now to FIG. 48 , there is illustrated the manner in which theoutput of the OAM generator 4006 may vary a signal by applying differentorbital angular momentums thereto. FIG. 48 illustrates helical phasefronts in which the Poynting vector is no longer parallel to the beamaxis and thus has an orbital angular momentum applied thereto. In anyfixed radius within the beam, the Poynting vector follows a spiraltrajectory around the axis. Rows are labeled by l, the orbital angularmomentum quantum number, L=lℏ is the beams orbital angular momentum perphoton within the output signal. For each l, the left column 4802 is thelight beam's instantaneous phase. The center column 4804 comprises theangular intensity profiles and the right column 4806 illustrates whatoccurs when such a beam interferes with a plane wave and produces aspiral intensity pattern. This is illustrated for orbital angularmomentums of −1, 0, 1, 2 and 3 within the various rows of FIG. 48 .

Referring now to FIG. 49 , there is illustrated an alternative manner inwhich the OAM generator 4006 may convert a Hermite-Gaussian beam outputfrom an emitter 4002 to a Laguerre-Gaussian beams having impartedtherein an orbital angular momentum using mode converters 4904 and aDove prism 4910. The Hermite-Gaussian mode plane waves 4902 are providedto a π/2 mode convertor 4904. The π/2 mode convertor 4904 produce beamsin the Laguerre-Gaussian modes 4906. The Laguerre-Gaussian modes beams4906 are applied to either a π mode convertor 4908 or a dove prism 4910that reverses the mode to create a reverse Laguerre-Gaussian mode signal4912.

Referring now to FIG. 50 , there is illustrated the manner in whichholograms within the OAM generator 4006 generate a twisted light beam. Ahologram 5002 can produce light beam 5004 and light beam 5006 havinghelical wave fronts and associated orbital angular momentum lh perphoton. The appropriate hologram 5002 can be calculated or generatedfrom the interference pattern between the desired beam form 5004, 5006and a plane wave 5008. The resulting holographic pattern within thehologram 5002 resembles a diffraction grating but has a l-prongeddislocation at the beam axis. When the hologram is illuminated with theplane wave 5008, the first-order diffracted beams 5004 and 5006 have thedesired helical wave fronts to provide the desired first ordereddiffracted beam display 5010.

Referring now to FIG. 51 , there is more particularly illustrated themanner in which the sample 4010 receives the input OAM twisted wave 4008provided from the OAM generator 4006 and provides an output OAM wave4012 having a particular OAM signature associated therewith that dependsupon the material or the concentration of a particular monitoredmaterial within the sample 4010. The sample 4010 may comprise any samplethat is under study and may be in a solid form, liquid form or gas form.The sample material 4010 that may be detected using the system describedherein may comprise a variety of different materials. As statedpreviously, the material may comprise liquids such as blood, water, oilor chemicals. The various types of carbon bondings such as C—H, C—O,C—P, C—S or C—N may be provided for detection. The system may alsodetect various types of bondings between carbon atoms such as a singlebond (methane or Isooctane), dual bond items (butadiene and benzene) ortriple bond carbon items such as acetylene.

The sample 4010 may include detectable items such as organic compoundsincluding carbohydrates, lipids (cylcerol and fatty acids), nucleicacids (C,H,O,N,P) (RNA and DNA) or various types of proteins such aspolyour of amino NH₂ and carboxyl COOH or aminos such as tryptophan,tyrosine and phenylalanine. Various chains within the samples 4010 mayalso be detected such as monomers, isomers and polymers. Enzymes such asATP and ADP within the samples may be detected. Substances produced orreleased by glands of the body may be in the sample and detected. Theseinclude items released by the exocrine glands via tube/ducts, endocrineglands released directly into blood samples or hormones. Various typesof glands that may have their secretions detected within a sample 4010include the hypothalamus, pineal and pituitary glands, the parathyroidand thyroid and thymus, the adrenal and pancreas glands of the torso andthe hormones released by the ovaries or testes of a male or female.

The sample 4010 may also be used for detecting various types ofbiochemical markers within the blood and urine of an individual such asmelanocytes and keratinocytes. The sample 4010 may include various partsof the body to detect defense substances therein. For example, withrespect to the skin, the sample 4010 may be used to detect carotenoids,vitamins, enzymes, b-carotene and lycopene. With respect to the eyepigment, the melanin/eumelanin, dihydroxyindole or carboxylic may bedetected. The system may also detect various types of materials withinthe body's biosynthetic pathways within the sample 4010 includinghemoglobin, myoglobin, cytochromes, and porphyrin molecules such asprotoporphyrin, coporphyrin, uroporphyrin and nematoporphyrin. Thesample 4010 may also contain various bacterias to be detected such aspropion bacterium, acnes. Also, various types of dental plaque bacteriamay be detected such as porphyromonos gingivitis, Prevotella intremediand Prevotella nigrescens. The sample 4010 may also be used for thedetection of glucose in insulin within a blood sample 4010. The sample4010 may also include amyloid-beta detection. Detection of amyloid-betawithin the sample may then be used for determinations of early onsetAlzheimer's. Higher levels of amyloid-beta may provide an indication ofthe early stages of Alzheimer's. The sample 4010 may comprise anymaterial that is desired to be detected that provides a unique OAM twistto a signal passing through the sample.

The orbital angular momentum within the beams provided within the sample4010 may be transferred from light to matter molecules depending uponthe rotation of the matter molecules. When a circularly polarized laserbeam with a helical wave front traps a molecule in an angular ring oflight around the beam axis, one can observe the transfer of both orbitaland spin angular momentum. The trapping is a form of optical tweezingaccomplished without mechanical constraints by the ring's intensitygradient. The orbital angular momentum transferred to the molecule makesit orbit around the beam axis as illustrated at 5202 of FIG. 52 . Thespin angular momentum sets the molecule spinning on its own axis asillustrated at 5204. This transference is also useful in inducingresonance within virus to destroy them.

The output OAM wave 4012 from the sample 4010 will have an orbitalangular momentum associated therewith that is different from the orbitalangular momentum provided on the input OAM wave 4008. The difference inthe output OAM wave 4012 will depend upon the material contained withinthe sample 4010 and the concentration of these materials within thesample 4010. Differing materials of differing concentration will haveunique orbital angular momentums associated therewith. Thus, byanalyzing the particular orbital angular momentum signature associatedwith the output OAM wave 4012, determinations may be made as to thematerials present within the sample 4010 and the concentration of thesematerials within the sample may also be determined.

Referring now to FIG. 53 , the matching module 4014 receives the outputorbital angular momentum wave 4012 from the sample 4010 that has aparticular signature associated therewith based upon the orbital angularmomentum imparted to the waves passing through the sample 4010. Thematching module 4014 amplifies the particular orbital angular momentumof interest in order to provide an amplified wave having the desiredorbital angular momentum of interest 4016 amplified. The matching module4014 may comprise a matching aperture that amplifies the detectionorbital angular momentum associated with a specific material orcharacteristic that is under study. The matching module 4014 may in oneembodiment comprise a holographic filter such as that described withrespect to FIG. 43 in order to amplify the desired orbital angularmomentum wave of interest. The matching module 4014 is established basedupon a specific material of interest that is trying to be detected bythe system. The matching module 4014 may comprise a fixed module usingholograms as illustrated in FIG. 43 or a tunable module in a mannersimilar to that discussed with respect to the OAM generation module4006. In this case, a number of different orbital angular momentumscould be amplified by the matching module in order to detect differingmaterials or differing concentrations of materials within the sample4010. Other examples of components for the matching module 4014 includethe use of quantum dots, nanomaterials or metamaterials in order toamplify any desired orbital angular momentum values within a receivedwave form from the sample 4010.

Referring now to FIG. 54 , the matching module 4014 rather than usingholographic images in order to amplify the desired orbital angularmomentum signals may use non-linear crystals in order to generate higherorbital angular momentum light beams. Using a non-linear crystal 5402, afirst harmonic orbital angular momentum beam 5404 may be applied to anon-linear crystal 5402. The non-linear crystal 5402 will create asecond order harmonic signal 5406.

Referring now to FIG. 55 , there is more particularly illustrated thedetector 4018 to which the amplified orbital angular momentum wave 4016from the matching circuit 4014 in order that the detector 4018 mayextract desired OAM measurements 5502. The detector 4018 receives theamplified OAM waves 4016 and detects and measures observable changeswithin the orbital angular momentum of the emitted waves due to thepresence of a particular material and the concentration of a particularmaterial under study within the sample 4010. The detector 4018 is ableto measure observable changes within the emitted amplified OAM wave 4016from the state of the input OAM wave 4008 applied to the sample 4010.The extracted OAM measurements 5502 are applied to the user interface4020. The detector 4018 includes an orbital angular momentum detector4204 for determining a profile of orbital angular momentum states of theorbital angular momentum within the orbital angular momentum signal 4016and a processor 4206 for determining the material within the sampleresponsive to the detected profile of the orbital angular momentumstates of the orbital angular momentum. The manner in which the detector4018 may detect differences within the orbital angular momentum is moreparticularly illustrates with respect to FIG. 56-58 .

FIG. 56 illustrates the difference in impact between spin angularpolarization and orbital angular polarization due to passing of a beamof light through a sample 5602. In sample 5602 a, there is illustratedthe manner in which spin angular polarization is altered responsive to abeam passing through the sample 5602 a. The polarization of a wavehaving a particular spin angular momentum 5604 passing through thesample 5602 a will rotate from a position 5604 to a new position 5606.The rotation occurs within the same plane of polarization. In a similarmanner, as illustrated with respect to sample 5602 b, an image appearsas illustrated generally at 5608 before it passes through the sample5602 b. Upon passing the image through the sample 5602 b the image willrotate from the position illustrated at 5610 to a rotated positionillustrated at 5612. The amount of rotation is dependent upon thepresence of the material being detected and the level of concentrationof the material being detected within the sample 5602. Thus, as can beseen with respect to the sample 5602 of FIG. 56 , both the spin angularpolarization and the orbital angular momentum will change based upon thepresence and concentration of materials within the sample 5602. Bymeasuring the amount of rotation of the image caused by the change inorbital angular momentum, the presence and concentration of a particularmaterial may be determined.

This overall process can be more particularly illustrated in FIG. 57 . Alight source 5702 shines a light beam through expanding optics 5704. Theexpanded light beam is applied through a metalab generated hologram 5706that imparts an orbital angular momentum to the beam. The twisted beamfrom the hologram 5706 is shined through a sample 5708 having aparticular length L. As mentioned previously, the sample 5708 may belocated in a container or in its naturally occurring state. This causesthe generation of a twisted beam on the output side of the sample 5708to create a number of detectable waves having various orbital angularmomentums 5710 associated therewith. The image 5712 associated with thelight beam that is applied to sample 5708 will rotate an angle φdepending upon the presence and concentration of the material within thesample 5708. The rotation φ of the image 5712 is different for eachvalue orbital angular momentum −l or +l. The change in rotation of theimage Δφ may be described according to the equation:Δφ=φ_(l)−φ⁻¹ =f(l,L,C)Where l is orbital angular momentum number, L is the path length of thesample and C is the concentration of the material being detected.

Thus, since the length of the sample L is known and the orbital angularmomentum may be determined using the process described herein, these twopieces of information may be able to calculate a concentration of thematerial within the provided sample.

The above equation may be utilized within the user interface moreparticularly illustrated in FIG. 58 . The user interface 4020 processesthe OAM measurements 5802 using an internal algorithm 5802 that providesfor the generation of material and/or concentration information 5804that may be displayed in some type of user display. The algorithm wouldin one embodiment utilize that equation described herein above in orderto determine the material and/or concentration based upon the length ofa sample and the detected variation in orbital angular momentum. Theprocess for calculating the material and/or concentration may be done ina laboratory setting where the information is transmitted wirelessly tothe lab or the user interface can be associated with a wearable deviceconnected to a meter or cell phone running an application on the cellphone connected via a local area network or wide area network to apersonal or public cloud. The user interface 5820 of the device caneither have a wired or wireless connection utilizing Bluetooth, ZigBeeor other wireless protocols.

Referring now to FIG. 59 , there is illustrated a particular example ofa block diagram of a particular apparatus for measuring the presence aconcentration of glucose using the orbital angular momentum of photonsof a light beam shined through a glucose sample. While the presentexample is with respect to the detection of glucose, one skilled in theart would realize that the example would be applicable to the detectionof the presence and concentration of any material. The process creates asecond-order harmonic with helical light beam using a non-linear crystalsuch as that described with respect to FIG. 54 . The emission module5902 generates plane electromagnetic waves that are provided to an OAMgeneration module 5904. The OAM generation module 5904 generates lightwaves having an orbital angular momentum applied thereto using hologramsto create a wave having an electromagnetic vortex. The OAM twisted wavesare applied to the sample 5906 that is under study in order to detectthe glucose and glucose concentration within a sample. A rotatedsignature exits the sample 5906 in the manner described previously withrespect to FIGS. 56-57 and is provided to the matching module 5908. Thematching module 5908 will amplify the orbital angular momentum such thatthe observed concentrations may be calculated from the orbital momentumof the signature of the glucose. These amplified signals are provided todetection module 5910 which measures the radius of the beam w(z) or therotation of the image provided to the sample via the light beam. Thisdetected information is provided to the user interface that includes asensor interface wired or wireless Bluetooth or ZigBee connection toenable the provision of the material to a reading meter or a user phonefor the display of concentration information with respect to the sample.In this manner concentrations of various types of material as describeherein may be determined utilizing the orbital angular momentumsignatures of the samples under study and the detection of thesematerials or their concentrations within the sample determine asdescribed.

Provided the orthogonality of Laguerre polynomials, Laguerre Gaussianbeams exhibiting orbital angular momentum (OAM) have been determined asa basis for spatial division multiplexing (SDM) in communicationapplications using for example a mux-demux optical element design. OAMbeams are also of interest in quantum informatics. OAM also enables theprobing of solutions of chiral and non-chiral molecules.

Referring now to FIG. 60 , there is illustrated a flow diagram foranalyzing intensity images taken by a camera. The intensity image hasapplied thereto threshold double precision amplitude to enable the ringto be clearly seen without extra pixels outside of the ring at step6002. Next at step 6001, both columns and rows are scanned along for theentire image. The peaks of the two largest hills and their locations aredetermined at step 6006. An ellipse is fit at step 4008 for all peaklocations found. Finally, at step 6010, a determination is made of themajor and minor axis of the ellipse, the focal point of the ellipse, thecentroid, eccentricity and orientation of the ellipse.

FIG. 61 illustrates an ellipse fitting algorithm flowchart. The X and Ypixel locations are input at step 6102 for all peaks that are found. Aninitial guess is provided at step 6104 for the conic equationparameters. The conic equation parameters comprise parameters A, B, C, Dand E for the equation Ax²+By²+Cx+Dy+E=0. The conjugate gradientalgorithm is used at step 6106 to find conic equation parameters thatprovide an optimal fit. An orientation of the ellipse is determined atstep 6108 and moved to determine the major and minor axis. Thedetermination of step 6108 is determined according to the equation

$\varnothing = {\frac{1}{2}\tan^{- 1}\frac{B}{C - A}}$The ellipse orientation is returned at step 6110 to determine thecentral point of the ellipse. Finally, at step 6112, a determination ismade if the conic equation represents an ellipse. For an ellipseparameters A and B will exist and have the same sign but will not beequal. Based upon this analysis it is been determined that lateral shiftof up to 1 mm can cause significant changes in the measured eccentricitydue to clipping of up to 0.2.Fractional OAM Signals

Molecular spectroscopy using OAM twisted beams can leverage fractionalOAM states as a molecular signature along with other intensitysignatures (i.e. eccentricity, shift of center of mass and rotation ofthe elliptical intensity) as well as phase signatures (i.e. changes inthe phase of the scattered beam) and specific formation of publicitydistributed spectrum. The method of optical orientation of electronicsbeen by circularly polarized photons has been heavily used to study spinangular momentum in solid state materials. The process relies onspin-orbit coupling to transfer angular momentum from the spin ofprotons to the spin of electrons and has been Incorporated intopump-probe Kerr and Faraday rotation experiments to study the dynamicsof optically excited spends. By enabling the study is spin decoherence,transport and interactions, this strategy has played a role in thedevelopment of semiconductor spintronics.

The proposed spectroscopy technique focuses instead on localized orbitalangular momentum (OAM) and solids. Specifically, one can distinguishbetween delocalized OAM associated with the envelope wave function whichmay be macroscopic in spatial extent, and local OAM associated withatomic sites, which typically is incorporated into the effect of spinand associated electronic states. The former type of angular momentum isa fundamental interest to orbital fleet coherent systems, for example,quantum Hall layers, superconductors and topological insulators.Techniques to study non-equilibrium delocalized OAM in these and othersystems create opportunities to improve understanding of scattering andquantum coherence of chiral electronic states, with potentialimplications for materials discovery.

The interaction of light with glucose in beta amyloid and thespectroscopy applications of OAM with respect to these. Additionally,the generation of Rahman sideband carrying OAM, OAM using a pleasantMonica lens, the study of optically coherent OAM in excite ions usingfor wave mixing in the application of linearly polarized light to createa 2-D pleasant Monica analog to OAM light in patterned sin metallicfilm, and the possibility of OAM light producing spin polarized votetill electronics for efficient semiconductors may also find applicationin these techniques.

Referring now to FIG. 62 , one manner for using nested fractional OAMstates to alleviate the problems associated with integer OAM states andto enable the use of stable states of fractional OAM for similarpurposes as those described herein above. In this case the input signals6202 are provided to fractional OAM generation circuitry 6204. Thefractional OAM generation circuitry 6204 generates output signals 6206having fractional orthogonal states which may then be further applied ordetected as discussed herein.

The orbital angular momentum of light beams is a consequence of theirazimuthal phase structure. Light beams have a phase factor exp(imφ),where m is an integer and φp is the azimuthal angle, and carry orbitalangular momentum (OAM) of mℏ per photon along the beam axis. These lightbeams can be generated in the laboratory by optical devices, such asspiral phase plates or holograms, which manipulate the phase of thebeam. In cases where such a device generates a light beam with aninteger value of m, the resulting phase structure has the form of |m|intertwined helices of equal phase. For integer values of m, the chosenheight of the phase step generated by the optical device is equal to themean value of the OAM in the resulting beam.

Recently, spiral phase steps with fractional step height as well asspatial holograms have been used to generate light beams with fractionalOAM states. In these implementations, the generating optical deviceimposes a phase change of exp(iMφ) where M is not restricted to integervalues. The phase structure of such beams shows a far more complexpattern. A series of optical vortices with alternating charge is createdin a dark line across the direction of the phase discontinuity imprintedby the optical device. In order to obtain the mean value of the orbitalangular momentum of these beams, one has to average over the vortexpattern. This mean value coincides with the phase step only for theinteger and half integer values. There are certainly more connectionsbetween optics and quantum theory to represent beams with fractional OAMas quantum states.

The theoretical description of light modes with fractional OAM is basedon the generating optical device. For integer OAM values, a theoreticaldescription may exist which provides the way to treat the angle itselfas quantum mechanical Hermitian operator. The description can providethe underlying theory for a secure quantum communication system and giveform to the uncertainty relation for angle and angular momentum. Thetheory may be generalized for fractional values of M thereby creating aquantum mechanical description of fractional OAM. Such a rigorousformulation is of particular interest is the use of half integer spiralphase plates have been used to study high dimensional entanglement.Fractional OAM states are characterized not only by the height of thephase step, but also by the orientation of the phase dislocation α. Forhalf odd integer values of M, M mod l=½, states with the same M but a πdifference in α are orthogonal. In light of recent applications ofinteger OAM in quantum key distribution in the conversion of spin toorbital angular momentum in an optical medium, a rigorous formulation isimportant for possible applications of fractional OAM to quantumcommunication.

The component of the OAM in the propagation direction L_(z) and theazimuthal rotation angle form a pair of conjugate variables (just liketime-frequency or space-momentum). Unlike linear position and momentum,which are both defined on an unbound and continuous state space, thestate spaces for OAM and the rotation angle are different in nature. TheOAM eigenstates form a discrete set of states with m taking on allinteger values. Eigenstates of the angle operator are restricted to a 2πradian interval since it is physically impossible to distinguish betweenrotation angles differing by less than 2π radians. The properties of theangle operator are rigorously derived in an arbitrarily large, yetfinite state space of 2L+1 dimensions. This space is spanned by theangular momentum states |m

with m ranging from −L, −L+1, . . . , L. Accordingly, the 2π radianinterval [θ0, θ0+2π) is spanned by 2L+1 orthogonal angle states |θn

with θn=θ0+2πn/(2L+1). Here, θ₀ determines the starting point of theinterval and with it a particular angle operator φ^(θ). Only afterphysical results have been calculated within this state space is Lallowed to tend to infinity, which recovers the result of an infinitebut countable number of basis states for the OAM and a dense set ofangle states within a 2π radian interval.

A quantum state with fractional OAM is denoted by |M

, where M=m+μ and m is the integer part and μ∈[0, 1) is the fractionalpart. The state |M

is decomposed in angle states according to:

$\left. M \right\rangle = {\left( {{2L} + 1} \right)^{- \frac{1}{2}}{\sum\limits_{n = 0}^{2L}\;{{\exp\left( {{iM}\;\theta_{n}} \right)}\left. \theta_{n} \right\rangle}}}$$\left. M \right\rangle = {\left( {{2L} + 1} \right)^{- \frac{1}{2}}{\sum\limits_{n = 0}^{2L}\;{{\exp\left( {{im}\;\theta_{n}} \right)}\mspace{14mu}{\exp\left( {i\;{\mu\theta}_{n}} \right)}\left. \theta_{n} \right\rangle}}}$

It is important to note that α is bounded by 0≤α<2π, so that theorientation of the discontinuity is always understood as measured fromθ₀. With this construction the fractional state |M

can be written as:

$\left. {M(\alpha)} \right\rangle = {\left( {{2L} + 1} \right)^{- \frac{1}{2}}{\exp\left( {i\;{\mu\alpha}} \right)}{\sum\limits_{n = 0}^{2L}\;{{\exp\left( {{iM}\;\theta_{n}} \right)}\mspace{14mu}{\exp\left\lbrack {i\; 2{\pi\mu}\;{f_{\alpha}\left( \theta_{n} \right)}} \right\rbrack}\left. \theta_{n} \right\rangle}}}$

In integer-based OAM generation applications light beams may begenerated using a spiral phase plate. However, light beams generatedusing a spiral phase plate with a non-integer phase step are unstable onpropagation. However, one can generate light carrying fractional orbitalangular momentum beams not with a phase step of a spiral phase plate butby a synthesis of Laguerre-Gaussian modes. This may be accomplished asillustrated in FIG. 63 using a spatial light modulator 6302. Inputsignals 6304 are provided to the spatial light modulator 6302 and usedfor the generation of fractional OAM beams 6306. The spatial lightmodulator 6302 synthesizes Laguerre Gaussian modes rather than using aphase step of a spiral phase plate. By limiting the number of Gouyphases in the superposition, one can produce a light beam from the SLM6302 which is well characterized in terms of its propagation. Thestructural stability of these fractional OAM light beams from an SLMmake them ideal for communications using fractional OAM states.Additionally, as will be described herein below, the beams would beuseful for concentration measurements of various organic materials.

Using the spatial light modulator 6302, a light beam with fractional OAMmay be produced as a generic superposition of light modes with differentvalues of m. As illustrated in FIG. 64 , various Laguerre-Gaussian beammodes 6402 may have a superposition process 6404 applied thereto by thespatial light modulator 6302 in order to generate the fractional beamoutputs 6406. Using the correspondence between optics and quantumtheory, OAM can be represented as a quantum state. This quantum state6502 can be decomposed into a basis of integer OAM states 6504 asgenerally illustrated in FIG. 65 . The decomposition only determines theOAM index m which in a superposition of LG beams leaves the index forthe number of concentric rings unspecified. Therefore, one can make useof this flexibility to find a representation of a fractional OAM statein terms of superimposed LG beams with a minimal number of Gouy phasesto increase propagation stability. One can produce these beams using thespatial light modulator 6302 and study their propagation and vortexstructure. Light beams constructed in this manner are in excellentrealization of non-integer OAM states and are more stable on propagationand light emerging from fractional faced steps of a spiral phase plate.

Referring now to FIG. 66 , there is illustrated the manner in which anSLM may be programmed to provide fractional OAM beams. Rather than usingmultiple optical elements to generate each Laguerre Gaussian modeseparately a single SLM 6602 may be programmed with a hologram 6604 thatsets the phase structure 6606 and intensity structure 6608 forgenerating the superposition. A blazed grating 6610 is also included inthe hologram 6604 to separate angularly the first fractional order. Theformula for the resulting phase distribution of the hologram 6604 andrectilinear coordinates Φ(x,y)_(holo) is given by:Φ(x,y)_(holo)=[Φ(x,y)_(beam)+Φ(x,Λ)_(grating) mod2π−π]sinc²[(1−I(x,y)_(beam))π]+π

In this equation Φ(x,y) beam is the phase profile of the superpositionat the beam waist for z=0 and Φ(x,Λ) grating is the phase profile of theblazed grating which depends on the period of the grating Λ. Thetwo-phase distributions are added to modulo 2π and, after subtraction ofπ are multiplied by an intensity mask. In regions of low intensity, theintensity mask reduces the effect of the blazed grating 6610, which inturn leads to reduced intensity in the first diffraction order. Themapping between the phase depth and the desired intensity is not linearbut rather given by the trigonometric sinc function.

Referring now to FIG. 67 and FIG. 68 , there are illustrated the stepsnecessary to generate a hologram for producing a non-integer OAM beam.Initially, at step 6802 a carrier phase representing a blazed grating6702 is added to the phase 6704 of the superposition modulo 2π. Thiscombined phase 6706 is multiplied at step 6804 by an intensity mask 6708which takes account of the correct mapping between the phase depth anddiffraction intensity 3010. The resulting hologram 6712 at step 6806 isa hologram containing the required phase and intensity profiles for thedesired non-integer OAM beam.

The use of fractional OAM beams may be used in a number of fashions. Inone embodiment, as illustrated in FIG. 69 , fractional OAM beams may begenerated from a fractional OAM beam generator 6902. These fractionalOAM beams are then shown through a sample 6904 in a manner similar tothat discussed herein above. OAM spectroscopy detection circuitry 6906may then be used to detect certain OAM fraction state profiles caused bythe OAM beam shining through the sample 6904. Particular OAM fractionstates will have a particular fractional OAM state characteristic causedby the sample 6904. This process would work in the same manner as thatdescribed herein above. Fractional OAM beams can also be used togenerate different resonances in a virus such as Covid-19.

FIG. 70 illustrates one example of a OAM state profile that may be usedto identify a particular material within a sample. In this case, thehighest number of OAM states is illustrated at L=3. Additional statelevels are also illustrated at L=1.5; L=2.75; L=3.5 and L=4. Thisparticular OAM state profile would be uniquely associated with aparticular material and could be used to identify the material within asample when the profile was detected. The interaction of LaguerreGaussian light beams with glucose and beta amyloid have been the initialspectroscopy application of OAM to sample types.

The transfer of OAM between the acoustic and photonic modes, thegeneration of Raman side bands carrying OAM, OAM using a plasmonic lens,the study of optically coherent OAM in excitons using four-wave mixing,the application of linearly polarized light to create a 2-D plasmonicanalog to OAM light in a patterned thin metallic film and thepossibility of OAM light producing spin polarized photoelectrons forefficient semiconductors are other potential spectroscopy applications.

Other means of generation and detection of OAM state profiles may alsobe utilized. For example, a pump-probe magneto-orbital approach may beused. In this embodiment Laguerre-Gaussian optical pump pulses impartorbital angular momentum to the electronic states of a material andsubsequent dynamics are studied with femto second time resolution. Theexcitation uses vortex modes that distribute angular momentum over amacroscopic area determined by the spot size, and the optical probestudies the chiral imbalance of vortex modes reflected off of a sample.There will be transients that evolve on timescales distinctly differentfrom population and spin relaxation but with large lifetimes.

Multi-Parameter Spectroscopy

The following discussions with respect to spectroscopy describe thedetection and creation of vibrational modes, molecular vibrations, etc.with respect to various spectroscopic techniques. The creation anddetection of these vibrational modes, molecular vibrations, etc. arealso appliable to viruses to induce the molecular resonance therein todestroy and sterilize the viruses.

A further application of the OAM spectroscopy may be further refined byidentifying items using a number of different types of spectroscopy toprovide a more definitive analysis. Referring now to FIG. 71 , there isgenerally illustrated a multi-parameter spectroscopy system 7100. Aplurality of different spectroscopy parameters 7102 may be tracked andanalyzed individually. The group of parameters is then analyzed togetherusing multi-parameter spectroscopy analysis processor or system 7104 todetermine and identify a sample with output 7106. The differentspectroscopic techniques receive a light beam generated from a lightsource 7108, for example a laser, that has passed through a sample 7110that a material or concentration of material therein that is beingdetected. While the light source of FIG. 71 illustrates a single laserand light beam, multiple light sources may provide multiple light beams,or a single source may be used to provide multiple light beams. In oneexample, development of a single optical spectroscopy system to fullycharacterize the physical and electronic properties of small samples inreal time may be accomplished using the polarization, wavelength, andorbital angular momentum (OAM) of light. A polarized optical source isused to characterize the atomic and molecular structure of the sample.The wavelength of the source characterizes the atomic and molecularelectronic properties of the sample including their degree ofpolarizability. OAM properties of the source are principally used tocharacterize the molecular chirality, but such new techniques are notlimited to chiral molecules or samples and can be applied to non-chiralmolecules or samples. These three spectroscopy dimensions combine togreatly improve the process of identifying the composition of materials.Integrated into a compact handheld spectrometer, 3D or multi-parameterspectroscopy empowers consumers with numerous applications includinguseful real time chemical and biological information. Combined withother pump-probe spectroscopy techniques, 3D/multi-parameterspectroscopy promises new possibilities in ultrafast, highly selectivemolecular spectroscopy. While the following description discusses anumber of different spectroscopy techniques that may be implemented inmulti-parameter spectroscopy system 7100, it should be realized thatother spectroscopy techniques may be combined to provide themulti-spectroscopy analysis system of the present disclosure.

Optical Spectroscopy

Spectroscopy is the measurement of the interaction of light with variousmaterials. The light may either be absorbed or emitted by the material.By analyzing the amount of light absorbed or emitted, a materialscomposition and quantity may be determined.

Some of the light's energy is absorbed by the material. Light of a givenwavelength interacting with a material may be emitted at a differentwavelength. This occurs in phenomena like fluorescence, luminescence,and phosphorescence. The effect of light on a material depends on thewavelength and intensity of the light as well as its physicalinteraction with the molecules and atoms of the material such as virus.

A schematic of a spectrometer which makes relative measurements in theoptical spectral region of the electromagnetic spectrum uses light thatis spectrally dispersed by a dispersing element is shown in FIG. 72 . Inparticular, a device 7202, such as a monochromator, polychromator, orinterferometer, selects a specific wavelength from a light source 7204.This single-wavelength light interacts with a sample 7206. A detector7208 is used to measure the spectrum of light resulting from thisinteraction. A change in the absorbance or intensity of the resultinglight 7210 is measured as the detector 7208 sweeps across a range ofwavelengths. A range of different spectroscopic techniques, based onthese fundamental measurements, have been developed such as thosediscussed in A. Hind, “Agilent 101: An Introduction to OpticalSpectroscopy,” 2011.(http://www.agilent.com/labs/features/2011_101_spectroscopy.html) whichis incorporated herein by reference in its entirety. Here, attention isgiven to molecular spectroscopy techniques including infrared, Raman,terahertz, fluorescence, and orbital angular momentum spectroscopy.

Molecular Spectroscopy

Infrared Spectroscopy

Various types of molecular spectroscopy techniques may also be used inthe multi-parameter spectroscopy system. These techniques includeinfrared spectroscopy and others.

Infrared frequencies occur between the visible and microwave regions ofthe electromagnetic spectrum. The frequency, ν, measured in Hertz (Hz),and wavelength, λ, typically measured in centimeters (cm) are inverselyrelated according to the equations:

$v = {{\frac{c}{\lambda}\mspace{14mu}{and}\mspace{14mu}\lambda} = \frac{c}{v}}$where c is the speed of light (3×10¹⁰ cm/sec).

The energy of the light is related to λ and ν by

$E = {{hv} = \frac{hc}{\lambda}}$where h is Planck's constant (h=6.6×10⁻³⁴ J·s).

The infrared (IR) spectrum is divided into three regions: the near-,mid-, and far-IR. The mid IR region includes wavelengths between 3×10⁻⁴and 3×10⁻³ cm.

In the process of infrared spectroscopy, IR radiation is absorbed byorganic molecules. Molecular vibrations occur when the infrared energymatches the energy of specific molecular vibration modes. At thesefrequencies, photons are absorbed by the material while photons at otherfrequencies are transmitted through the material.

The IR spectrum of different materials typically includes uniquetransmittance, T, peaks and absorbance troughs occurring at differentfrequencies such as the measured IR spectrum of water vapor.

The absorbance, A, is related to the transmittance byA=log₁₀(1/T).

Each material exhibits a unique infrared spectral fingerprint, orsignature, determined by its unique molecular vibration modes whichpermit identification of the material's composition by IR spectroscopy.In the case of water vapor (FIG. 62 ), for example, the water moleculesabsorb energy within two narrow infrared wavelengths bands that appearas absorbance troughs 6202.

Molecular Vibrations

As described above one manner for inactivating viruses is by theinducement of molecular vibrations within the viruses using OAM andother techniques. Referring now to FIG. 73 , water molecules exhibit twotypes of molecular vibrations: stretching and bending. A molecule 7302consisting of n atoms 7308 has 3n degrees of freedom. In a nonlinearmolecule like water, three of these degrees are rotational, three aretranslational, and the remaining correspond to fundamental vibrations.In a linear molecule 7302, two degrees are rotational and three aretranslational. The net number of fundamental vibrations for nonlinearand linear molecules is therefore, 3n−6 and 3n−5, respectively.

For water vapor, there are two strong absorbance troughs occurring atapproximately 2.7 μm and 6.3 μm as a result of the two stretchingvibrational modes 7304 of water vapor and its bending mode 7306,respectively. In particular, the symmetric and asymmetric stretchingmodes 7304 absorb at frequencies in very close proximity to each other(2.734 μm and 2.662 μm, respectively) and appear as a single, broaderabsorbance band between the troughs.

Carbon dioxide, CO₂, exhibits two scissoring and bending vibrations5302, 5304 (FIG. 53 ) that are equivalent and therefore, have the samedegenerate frequency. This degeneracy appears in the infrared spectrumat λ=15 μm. The symmetrical stretching vibrational mode 5304 of CO₂ isinactive in the infrared because it doesn't perturb its molecular dipolemoment. However, the asymmetrical stretching vibration mode 5302 of CO₂does perturb the molecule's dipole moment and causes an absorbance inCO₂ at 4.3 μm.

Both molecular stretching and bending vibration modes of molecules(FIGS. 73 and 74 ) can be predicted to useful theoretical approximationusing simple classical mechanics models.

Stretching Vibrations

Similarly stretching vibrations may be used to induce vibrations withinviruses to destroy them. The stretching frequency of a molecular bondmay be approximated by Hooke's Law when treated as a simple classicalharmonic oscillator consisting of two equal masses bound by a spring

$v = {\frac{1}{2\pi}\sqrt{\frac{k}{m}}}$where k is the force constant of the spring and m is the mass of anatom.

In the classical harmonic oscillator, the energy depends on the extentto which the spring is stretched or compressed,E=½kx ² =hvwhere x is the displacement of the spring. The classical model ofHooke's Law, however, is inconsistent with the absorbance of energy bymolecules as it would suggest that energy of any frequency is absorbed.In real molecules, vibrational motion is quantized and appropriatelymodeled by the quantum mechanical expression,E _(n)=(n+½)hvwhere n is the principal quantum number (n=0, 1, 2, 3 . . . )characteristic of each permitted energy level.

The lowest energy level is E₀=½hv followed by E₁=3/2hv. Only transitionsto the next energy level are allowed according to the selection rule.Subsequently, molecules absorb photonic energy in integer increments ofhv. For photon absorption energies of 2hv or 3hv, however, the resultingabsorbance bands are called overtones of the infrared spectrum and areof lesser intensity than fundamental vibrational bands.

Energy of a harmonic oscillator as a function of the interatomicdistance is shown in FIG. 75 with an energy minimum occurs at the normalbond length 7502 (equivalent to a relaxed classical mechanical spring).As the interatomic distance increases the quantized energy levels 7504become more closely spaced and the energy reaches a maximum. The allowedtransitions, hv become smaller in magnitude which gives lower overtoneenergies than would otherwise be predicted using the simply harmonicoscillator theory depicted in FIG. 76 .

Though this mathematical framework represents a useful, if not simple,approximation, the vibrational activity between two atoms in a largemolecule cannot be isolated from the vibrational behavior of other atomsin the molecule. Vibrations of two bonds within a molecule may becoupled in such a manner that one contracts or expands while the othercontracts as in either asymmetrical or symmetrical stretchingvibrations. When this occurs different absorbance frequency bands areobserved instead of superimposed, or degenerate, bands as observed whentwo identical atoms in a bond vibrate with an identical force constant.

Infrared spectroscopy is used to identify material species by theirunique vibrational and rotational optical signatures. A complementaryspectroscopy technique, Raman spectroscopy is used to identify materialsby their unique light-scattering signatures as discussed in the nextsection.

Raman Spectroscopy

Since Raman spectroscopy is a technique used to characterize a materialby the amount of light it scatters. Raman spectroscopy complementsinfrared spectroscopy which instead measures the amount of lightabsorbed by a material. Raman and infrared spectroscopy may further beused in conjunctions with OAM and polarization spectroscopy to furtherimprove analysis results. When light interacts with matter, changes inthe dipole moment of its molecules yield infrared absorption bands whilechanges in their polarizability produce Raman bands. The sequence ofobserved energy bands arises from specific molecular vibrations whichcollectively produce a unique spectral signature indicative of each typeof molecule. Certain vibrational modes occurring in Raman spectroscopyare forbidden in infrared spectroscopy while other vibrational modes maybe observed using both techniques or a multi-parameter technique usingOAM. When these latter modes are common to both techniques, theirintensities differ significantly.

The most frequent interaction of photons with molecules results inRayleigh scattering in which photons are elastically scattered as theresult of excited electrons that decay to their original energy level.Consequently, Rayleigh scattered photons have the same energy asincident photons.

With the discovery of inelastic photonic scattering phenomena in 1928 byC. V. Raman and K. S. Krishnan, Raman spectroscopy was established as apractical chemical analysis method useful to characterize a wide varietyof chemical species including solid, liquid, and gaseous samples. Solidcrystal lattice vibrations are typically active in Raman spectroscopyand their spectra appear in polymeric and semiconductor samples. Gaseoussamples exhibit rotational structures that may be characterized byvibrational transitions.

Approximately one percent of incident photons scatter inelastically andyield lower energy photons. Raman scattering results from changes in thevibrational, rotational, or electronic energy of a molecule. Thevibrational energy of the scattering molecule is equivalent to thedifference between incident and Raman scattered photons. When anincident photon interacts with the electric dipole of a molecule, thisform of vibronic spectroscopy is often classically viewed as aperturbation of the molecule's electric field. Quantum mechanically,however, the scattering event is described as an excitation to a virtualenergy state lower in energy than a real electronic transition withnearly coincident decay and change in vibrational energy. Suchspectroscopy can work in conjunction with incident photons that carryOAM. In Raman spectroscopy, incident photons excite electrons to adifferent final energy level than its original energy level (FIG. 77 ).

Since the intensity of Raman scattering is low, heat produced by thedissipation of vibrational energy does not yield an appreciable rise inmaterial temperature. Such Raman spectroscopy can work in conjunctionwith incident photons that carry OAM. At room temperature, thepopulation of vibrationally excited states is small. Stokes-shiftedscattering events shown in FIG. 77 are typically observed in Ramanspectroscopy since at room temperature the excited vibrational statesare low, and the electron originates in the ground state. The inelasticRaman scattered photon 7702 has lower energy than the incident photon7704 as the electron decays to an energy level 7706 higher than theoriginal ground state 7708. Anti-Stokes shifted scattering events 7710result from a small fraction of molecules originally in vibrationallyexcited states (FIG. 77 ) which leave them in the ground state 7712 andresults in Raman scattered photons with higher energy. At roomtemperature, anti-Stokes shifted Raman spectra are always weaker thanStokes-shifted spectrum since the Stokes and anti-Stokes spectra containthe same frequency information. Most Raman spectroscopy focusesexclusively on Stokes-shifted scattering phenomena for this reason.

The force constant by which the vibrational mode energy may be modeledis affected by molecular structure including atomic mass, molecularspecies, bond order, and the geometric arrangement of molecules.However, Raman scattering occurs when the polarizability of moleculesmay be affected.

The polarizability, α, of a molecule appears as a proportionalityconstant between the electric field and the induced dipole moment,P=αE.

The induced dipole scatters a photon at the frequency of the incidentphoton (Rayleigh scattering). Molecular vibration, however, may changethe polarizability and give rise to inelastic Raman scattering ofphotons. Changes in polarizability may be expressed by

$\frac{\partial\alpha}{\partial Q} \neq 0$where Q is in a direction normal to the vibration, and is considered aselection rule for Raman-active vibrations.

Raman-active vibrations are non-existent in the infrared for moleculeshaving a center of symmetry while the existence of a perturbed symmetrycenter (e.g. permanent dipole moment) indicates the absence ofinfrared-active vibrations.

The intensity of a Raman band is proportional to the square of thespatial change of polarizability, or the induced dipole moment,

$I_{Raman} \propto {\left( \frac{\partial\alpha}{\partial Q} \right)^{2}.}$

Hence, incident photons that slightly induce a dipole moment will yielda Raman band with a very small intensity. Stronger Raman scatteringsystems are those with higher values of α such as molecules havingdouble carbon bonds which exhibit more broadly distributed electronssusceptible to polarization. Subsequently, the range of chemicalconcentrations measurable by Raman spectroscopy is considerably widegiven that the scattering intensity is directly proportional toconcentration.

Raman spectroscopy exhibits several advantages over other spectroscopytechniques. Raman bands exhibit good signal-to-noise ratios owing to itsdetection of fundamental vibrational modes. Hence, the Raman signatureof measured samples is typically more pronounced and definitive.

Raman spectroscopy is more useful for analyzing aqueous solutions thaninfrared spectroscopy since the Raman spectrum of water is weak andunobtrusive while the infrared spectrum of water is very strong and morecomplex. In organic and inorganic chemistries, the existence of covalentbonds yields a unique Raman signature. A Raman spectroscopy setup onlyrequires an appropriate laser source incident on a material and adetector to collect scattered photons which minimizes the need forelaborate sample preparation. Raman spectroscopy is non-destructive asthe material is merely illuminated with a laser. Because the Ramaneffect is weak, the efficiency and optimization of a Raman spectroscopyinstrument is critically important to providing measurements of theslightest molecular concentrations within the shortest possible time.

Spontaneous Raman Spectroscopy

The intensity of spontaneous Raman scattering is linearly dependent onthe incident intensity of light but of several orders of magnitude lessintense. Treating the light-matter interaction quantum mechanically, thetotal Hamiltonian may be expressed in terms of the energy associatedwith the vibrational modes of the molecule, H_(v), the light, H_(γ), andtheir interaction, H_(vγ),H=H _(v) +H _(γ) +H _(vγ).

In this framework

$H_{v} = {\frac{1}{2m}\left( {p^{2} + {\omega_{0}^{2}q^{2}}} \right)}$with vibrational frequency ω₀ and the normal mode amplitude q which maybe expressed in terms of creation and annihilation operators of themolecular vibrations,

$q = {\sqrt{\frac{2{\pi\hslash}}{8\pi^{2}{\mu\omega}_{0}}}\left\lbrack {b^{\dagger} + b} \right\rbrack}$with the electric dipole moment μ. This leavesH _(v)=ℏω₀(b ^(†) b+½).

Using creation and annihilation operators for light, a^(†) and a, fieldquantization is obtained,

$E_{\lambda} = {\sqrt{\frac{2\pi\;{hv}_{L}}{ɛ\; V_{int}}}{\sum\limits_{k_{\lambda}}{e_{k_{\lambda}}{i\left\lbrack {{\alpha\; k_{\lambda}^{\dagger}} - {\alpha\; k_{\lambda}}} \right\rbrack}}}}$where e_(k) _(λ) is the field polarization unit vector field and V_(int)the interaction volume. The Hamiltonian for the light is then

$H_{\gamma} = {\sum\limits_{k_{\lambda}}{{{\hslash\omega}_{k_{\lambda}}\left( {{a_{k_{\lambda}}^{\dagger}a_{k_{\lambda}}} + {1\text{/}2}} \right)}.}}$

Using the first order perturbation of the electric dipole approximationthe interaction Hamiltonian may be obtained in terms of the molecule'spolarizability, α,

$\begin{matrix}{H_{int} = {E \cdot \alpha \cdot E}} \\{= {{E \cdot \alpha_{0} \cdot E} + {\left( \frac{\partial\alpha}{\partial q} \right)_{0}{E \cdot q \cdot E}} + \cdots}}\end{matrix}$

within the local coordinate system, q. The first term characterizesRayleigh scattering. The remaining first order Raman scattering term isneeded to characterize spontaneous Raman scattering including thecoherent laser field, E_(L), in addition to the Stokes and anti-Stokesfields, E_(S) and E_(AS), respectively. Substituting q and E_(γ) intothis expression yields

$H_{int} = {{H_{\gamma\; S} + H_{\gamma\;{AS}}} \sim {{\left( \frac{\partial\alpha}{\partial q} \right)_{0}{\sum\limits_{k_{S}k_{L}}{\sqrt{\frac{\left( {2\omega_{L}\omega_{S}} \right)}{\omega_{0}}}\left( {e_{k_{L}} \cdot e_{k_{S}}} \right)\left( {{a_{k_{S}}^{\dagger}b^{\dagger}a_{k_{L}}} + {a_{k_{S}}{ba}_{k_{L}}^{\dagger}}} \right){\delta\left( {k_{L} - k_{S} - k_{v}} \right)}}}} + {\left( \frac{\partial\alpha}{\partial q} \right)_{0}{\sum\limits_{k_{AS}k_{L}}{\sqrt{\frac{\left( {2\omega_{L}\omega_{AS}} \right)}{\omega_{0}}}\left( {e_{k_{L}} \cdot e_{k_{AS}}} \right)\left( {{a_{k_{S}}^{\dagger}{ba}_{k_{L}}} + {a_{k_{AS}}b^{\dagger}a_{k_{L}}^{\dagger}}} \right){\delta\left( {k_{L} - k_{AS} + k_{v}} \right)}}}}}}$where H_(γS) and H_(γAS) are the interaction Hamiltonians of the Stokesand anti-Stokes branches, respectively.

The steady state transition rate between the initial, |i

, and final, |f

states is given according to Fermi's golden rule,

$W_{i\rightarrow f} = {\frac{2\pi}{\hslash}{\left\langle {f{H_{int}}i} \right\rangle }^{2}{{\rho\left( {\hslash\omega}_{f} \right)}.}}$

In the simple harmonic oscillator picture, the eigenstates, |n_(v)

with excitation quanta n_(v) are acted upon by creation and annihilationoperators to yield the Stokes and anti-Stokes transition ratesW _(n) _(v) →n _(v)+1, and W _(n) _(v) →n _(v)−1˜n _(v).

Hence, it is easy to determine n_(v) from the Raman signal intensitygiven a linear dependence.

Raman intensities from each vibrational level are used to identifyunique vibrational molecular modes and characterize the material'scomposition.

The integrated anti-Stokes intensity of a Raman mode is proportional tothe average vibrational quantum number of the mode,

n_(v)

,

$I_{AS} = {{A\left( \frac{E_{R}}{{hv}_{R}} \right)} = {A\left\langle n_{v} \right\rangle}}$where A is the Raman cross section. Normalizing I_(AS) with respect tothe room temperature Stokes signal of the same mode in addition to usingthe Boltzmann distribution,

$\left\langle n_{v} \right\rangle_{0} = {\frac{E_{R}^{0}}{hv_{R}} = \frac{1}{e^{\frac{hv_{R}}{kT_{0}}} - 1}}$where E_(R) ⁰ is the room temperature (T₀) energy of the Raman mode.Generally, hv_(R)>>kT₀. so

n_(v)

₀=0, and the normalized anti-Stokes signal is approximately

n_(v)

,

${I_{norm} \equiv \frac{I_{AS}}{I_{R}^{0}}} = {\frac{A\left\langle n_{v} \right\rangle}{A\left( {1 + \left\langle n_{v} \right\rangle_{0}} \right)} \approx {\left\langle n_{v} \right\rangle.}}$

By comparing the normalized scattering intensities associated withdifferent vibrational moved, the distribution of energy over differentmolecular modes after infrared excitation may be obtained.

Stimulated Raman Spectroscopy

Stimulated Raman intensity is nonlinearly dependent on the incidentintensity of photons but of similar magnitude. Inelastic scattering of aphoton with an optical phonon originating from a finite response time ofthe third order nonlinear polarization of a material is characteristicof Raman scattering. Monochromatic light propagating in an opticalmaterial yields spontaneous Raman scattering in which some photons aretransitioned to new frequencies. The polarization of scattered photonsmay be parallel or orthogonal if the pump beam is linearly polarized.Stimulated Raman scattering occurs when the scattering intensity ofphotons at shifted frequencies is enhanced by existing photons alreadypresent at these shifted frequencies. Consequently, in stimulated Ramanscattering, a coincident photon at a downshifted frequency receives again which may be exploited in Raman amplifiers, for example, orusefully employed in molecular spectroscopy.

Raman amplification became a mature technology with the availability ofsufficiently high-power pump lasers.

Within a classical electromagnetic framework, the stimulated Ramanscattered signal intensity increases proportionally with the pump andsignal intensities

$\frac{{dI}_{s}}{dz} = {g_{R}I_{P}I_{S}}$

and the Raman-gain coefficient, g_(R), which is related to thespontaneous Raman scattering cross section. Hence, the probability ofRaman scattering is directly related to the photon density in the pumpwave and the Raman cross section.

The Stokes and pump waves must overlap spatially and temporally togenerate stimulated emission. Since, the Raman process involvesvibrational modes of molecules within a material; its intensity spectrumdetermines the material composition. In amorphous materials, forexample, the vibrational energy levels tend to merge, and form bands andthe pump frequency may differ from the Stokes frequency over a widerange. In crystalline materials, however, the intensity peaks tend to bewell-separated as they have narrow bandwidths.

The coupled wave equations for forward Raman scattering include

$\frac{{dI}_{S}}{dz} = {{g_{R}I_{p}I_{S}} - {\alpha_{S}I_{S}}}$for Stokes intensities with α_(S) the Stokes attenuation coefficient,and

$\frac{{dI}_{P}}{dz} = {{{- \frac{\omega_{P}}{\omega_{S}}}g_{R}I_{p}I_{s}} - {\alpha_{P}I_{P}}}$for pump wave intensities where ω_(P) and ω_(S) are pump and Stokesfrequencies, respectively. For backward scattering,dI_(S)/dz→−dI_(S)/dz. In the absence of loss, these expressions reduceto

${\frac{d}{dz}\left( {\frac{I_{S}}{\omega_{s}} + \frac{I_{P}}{\omega_{P}}} \right)} = 0$

which embodies the conservation of photon number in Stokes and pumpwaves during stimulated Raman scattering processes.

Stimulated scattering intensity increases when the stimulated gainexceeds the linear loss which is the source of the threshold power whichmust be overcome to initiate stimulated Raman scattering. In a materialsystem in which forward and backward scattering occurs, a beat frequencydrives molecular oscillations responsible for increasing the scatteredwave amplitude. In turn, the increasing wave amplitude enhances themolecular oscillations as part of a positive feedback loop that resultsin the stimulated Raman scattering effect. For forward scatteringprocesses, the pump depletion term is removed,

$\frac{{dI}_{P}}{dz} = {{- \alpha_{p}}{I_{P}.}}$

Solving this equation yields I_(P)(z)=I₀e^(−α) ^(P) ^(z) giving thestimulated Stokes scattering intensityI _(S)(L)=I _(S)(0)e ^(g) ^(R) ^(I) ⁰ ^(L) ^(eff) ^(−α) ^(P) ^(L)

where the effective optical path length is given by

$L_{eff} = {\frac{1 - e^{{- \alpha_{p}}L}}{\alpha_{P}}.}$

Stimulated Raman scattering intensifies from scattering events occurringthroughout the optical path length in the material, making it a usefulmolecular spectroscopy technology.

Resonance Raman Spectroscopy

The Raman effect in classical Raman spectroscopy depends only on thefrequency of incident light with scattered intensity dependence on ν₀ ⁴as discussed earlier. If the vibrational mode of a molecular absorptiontransition precisely matches the energy of incident light, the observedscattered intensity may be as intense as ˜ν₀ ⁶. This resonance Ramaneffect permits highly sensitive spectroscopic discrimination of amolecular species within a complex material medium such as chromophoreswithin proteins embedded in a biological membrane.

In resonance Raman spectroscopy, only a small fraction of molecularvibrational modes are enhanced. In the simplest scenario, only oneelectronic state may be resonant. In this case, the resonant Ramansignal is the result of nuclear motion resulting from distortions of themolecule while transitioning between the ground state and the excitedstate in which resonance is induced by incident light.

The functional component of most biological chromophores consists ofatoms conjugated with the particular electronic transition to whichresonance Raman spectroscopy is selectively sensitive. The frequency ofmeasured resonance Raman bands yields information about the vibrationalstructure of the electronic states involved in the transition used forinducing the resonance. The scattering intensities provide informationabout the nature of mode coupling with the electronic transition.

Raman Effect in Vortex Light

A molecule in vibronic state m subjected to a plane-polarized incidentlight of frequency ν₀ and intensity I₀ is perturbed into a new vibronicstate n. This interaction causes the frequency of light to shift byν_(mn)=ν_(m)−ν_(n) and scatter with a frequency ν₀+ν_(mn) through asolid angle 4π. The scattering intensity during the transition from m ton is given by

$I_{mn} = {\frac{2^{6}\pi^{4}}{3c^{3}}\left( {v_{0} + v_{mn}} \right)^{4}{{\mathfrak{E}}_{mn}}^{2}}$in which the amplitude

_mn of the electric field is given by

${\mathfrak{E}}_{mn} = {\frac{1}{h}{\sum\limits_{r}\left( {\frac{M_{m}\left( {M_{mr}{\mathfrak{U}}} \right)}{v_{rm} - v_{0}} + \frac{M_{mr}\left( {M_{rn}{\mathfrak{U}}} \right)}{v_{rn} + v_{0}}} \right)}}$where, m, r and n are quantum numbers of the initial, intermediate andfinal energy states E_(m), E_(r), E_(n), respectively.

Between the amplitude

of the electric field strength

=

e ^(−2πiν) ⁰ ^(t) +

*e ^(2πiν) ⁰ ^(t)and its amplitude

_(mn) associated with the shifted scattered radiation induced torque,

M_(mn) = 𝔈_(mn)e^(−2πi(v₀ + v_(mn))t) + 𝔈_(mn)^(*)e^(2πi(v₀ + v_(mn))t)is a tensor relation that may be expressed in terms of scattering tensorA_(mn)=(α_(ρσ))_(mn) the form:

_(mn) =A _(mn)

or in component representation,

$\left( {\mathfrak{E}}_{\rho} \right)_{mn} = {\sum\limits_{\sigma}{\left( \alpha_{\rho\sigma} \right)_{mn}{\mathfrak{U}}_{\sigma}}}$while the scattering tensor A_(mn) may be expressed as

${A_{mn} = {\frac{1}{h}{\sum\limits_{r}\left( {\frac{M_{rn}M_{mr}}{v_{rm} - v_{o}} + \frac{M_{mr}M_{rn}}{v_{rn} + v_{0}}} \right)}}},$

Since

_(mn) written in terms dyadic components of the tensor A_(mn) includesM_(rn)M_(mr), each path matrix element of the polarizability tensor, α,for a transition from m to n, may be written in terms of intermediatevibronic states

${\left( \alpha_{\rho\sigma} \right)_{mn} = {\frac{1}{2{\pi\hslash}}{\sum\limits_{r}\left( {\frac{\left( M_{\rho} \right)_{rn}\left( M_{\sigma} \right)_{mr}}{v_{rm} - v_{0}} + \frac{\left( M_{\rho} \right)_{mr}\left( M_{\sigma} \right)_{rn}}{v_{rn} + v_{0}}} \right)}}},$

Where (M_(ρ))_(mn) is the transition matrix between vibrational levels mand n in the presence of the radiation operator {circumflex over(m)}_(ρ),(M _(ρ))_(mn)=·Ψ_(r) *{circumflex over (m)} _(ρ)Ψ_(m) dτ

Herein, (

)_(rn)(M_(σ))_(mr) are ordinary products of scalar vector components (

)_(rn) and (M_(σ))_(mr) of a unit vector a_(σ). In the three mutuallyperpendicular directions spatially fixed

, σ=1, 2, 3 as follows:

${{\mathfrak{E}}_{mn}}^{2} = {{\sum\limits_{\rho}\left( {\mathfrak{E}}_{\ell} \right)_{mn}^{2}} = {{\sum\limits_{\rho}{{\sum\limits_{\sigma}{\left( \alpha_{\rho\sigma} \right)_{mn}{\mathfrak{U}}_{\sigma}}}}^{2}} = {A^{2}{\sum\limits_{\rho}{{\sum\limits_{\sigma}{\left( \alpha_{\rho\sigma} \right)_{mn}{\mathfrak{a}}_{\sigma}}}}^{2}}}}}$

With an incident intensity, I₀=(c/2π)A², then,

$I_{mn} = {{\frac{2^{6}\pi^{4}A^{2}}{3c^{3}}\left( {v_{0} + v_{mn}} \right)^{4}{\sum\limits_{\rho}{{\sum\limits_{\sigma}{\left( \alpha_{\rho\alpha} \right)_{mn}{\mathfrak{a}}_{\sigma}}}}^{2}}} = {\frac{2^{7}\pi^{5}}{3c^{3}}{I_{0}\left( {v_{0} + v_{mn}} \right)}^{4}{\sum\limits_{\rho}{{{\sum\limits_{\sigma}{\left( \alpha_{\rho\sigma} \right)_{mn}{\mathfrak{a}}_{\sigma}}}}^{2}.}}}}$

The total scattering intensity is therefore dependent on the state ofpolarization of the exciting light. By averaging over all positions ofa, or averaging over all modes of the scattering molecule at a fixedincident wave direction and polarization,

$\overset{\_}{{{\sum\limits_{\sigma}{\left( \alpha_{\rho\sigma} \right)_{mn}{\mathfrak{a}}_{\sigma}}}}^{2}} = {\frac{1}{3}{\sum\limits_{\sigma}{{\left( \alpha_{\rho\sigma} \right)}^{2}.}}}$

Finally, for an electron transition from m→n per molecule an averagetotal intensity of the scattered radiation is obtained

$I_{mn} = {\frac{2^{7}\pi^{5}}{3^{2}c^{4}}{I_{0}\left( {v_{0} + v_{mn}} \right)}^{4}{\sum\limits_{\rho,\sigma}{\left( \alpha_{\rho\alpha} \right)_{mn}}^{2}}}$in which ρ=x, y, z and σ=x′, y′, z′ are independently the fixedcoordinate systems of the molecule for incident and scattered photons,respectively.Selection Rules for Raman Effect Using Vortex Light

Of interest to studies of the Raman effect using vortex light is aparticular set of solutions of Maxwell's equations in a paraxialapproximation. Laguerre-Gaussian functions may mathematicallycharacterize a beam of vortex light in terms of generalized Laguerrepolynomials,

(x) with a Gaussian envelope. In the Lorentz-gauge, the vector potentialof a Laguerre-Gaussian beam is:

$A_{\ell,p} = {{A_{0}\left( {{\alpha{\hat{e}}_{x}} + {\beta{\hat{e}}_{y}}} \right)}\sqrt{\frac{2{p!}}{{\pi\left( {{\ell } + p} \right)}!}}\frac{w_{0}}{w(z)}{L_{LP}^{\ell }\left( \frac{2\rho^{2}}{w^{2}(z)} \right)}\left( \frac{\sqrt{2}\rho}{w(z)} \right)^{\ell }e^{{i\;{\ell\phi}} - {i\;\omega\; t} + {ikz}}}$in a (ρ, ϕ, z) coordinate system in which w(z) is the beam waist(radius) at which the radial field amplitude goes to 1/e. Forsimplicity, only p=0 is typically chosen. In the dipole approximation,the term, e^(ikz) is negligible, so the radiation operator of aLaguerre-Gaussian beam may be expressed as

${\hat{m}}_{\rho} = {{\left\lbrack {{A_{0}\left( {{\alpha\;{\hat{e}}_{x}} + {\beta\;{\hat{e}}_{y}}} \right)}\sqrt{\frac{1}{{\pi!}{{\ell }!}}}\frac{w_{0}}{w(z)}{L_{0}^{i\;\hslash}\left( \frac{2\rho^{2}}{w^{2{(z)}}} \right)}\left( \frac{\sqrt{2}\rho}{w(z)} \right)^{i\;\hslash}e^{{{il}\;\phi} - {i\;\omega\; t}}} \right\rbrack \cdot p} + {c \cdot c}}$

Here, e^(iωt) is associated with photon emission and e^(−iωt) isassociated with photon absorption.

The following generalized framework for developing a set of selectionrules to measure unique OAM Raman signatures of different materialsapplies to the intensity profiles associated with both stimulated andspontaneous Raman spectroscopy.

The relationship among irreducible representations of the phonon, theincident photon, and the scattering photon, Γ_(α), Γ_(ρ), and Γ_(σ),required to ensure non-vanishing matrix elements of

_(,p) isΓ_(α)

Γ_(ρ)

Γ_(σ)

Γ₁such that h_(e,s) ^(α), (M_(ρ))_(g,e), and (M_(σ))_(g,s) are non-zero.Introducing, the Raman tensor P_(αβγδ)(Γ_(j) ^(σ)) having index Γ_(j)^(σ) to denote the jth branch of the σth phonon to replace the singleindex α, we similarly replace the incident photon index, ρ, with (α, β)and the scattered photon index, σ with (γ, δ).

As the interaction of light with matter in Raman scattering processesleaves the orbital angular momentum of photons unperturbed the incidentand scattered photons may be expressed in the following respectiveforms,(ρ·ϵ₁)

and (ρ·ϵ_(S))

.

Then P_(αβγδ)(Γ_(j) ^(σ)) may be determined by the Clebsch-Gordancoefficients for all three representationsP _(z,ϵ) _(S) _(,ϵ) _(I) _(z)(Γ_(j) ^(σ))=(ρ·ϵ_(S))

⊗(ρ·ϵ_(I))

⊗ϕ_(σ) ^(j)

For crystalline materials, the special case of forward scatteringreduces 3×3 Raman tensors to 2×2. In this case, the Raman tensors for

≥2 excitations all have the same form. So from symmetry considerations,the

-dependence vanishes for

≥2. Since the constants a, b, c, d, and e depend on

and the symmetry of the crystal, non-zero OAM yields a Γ₂ phonon for

≥2 photon excitation and decouples the two Raman tensors for the Γ₃phonon for

≥1 photon excitation.

OAM Raman spectroscopy exhibits the capacity to characterize the atomicand molecular composition of a crystalline material. More complicatedselection rules are needed to fully obtain an OAM Raman signature ofchiral materials which present their own unique atomic and molecularsymmetry properties.

In the highly symmetric case of crystalline materials, for example, theapproach is rather straightforward. Given a periodic lattice potential,electrons in crystal solids may be expressed as Bloch wavesψ_(n,k)(r)=e ^(ik·r) u _(nk)(r)such that the electron transition moment connecting the ground state,ψ_(g,k), to the excited state, ψ_(e,k), may be written

$M_{g,e} = {\sum\limits_{k}{\int{{{\psi_{e,k}^{*}(r)}\left\lbrack {{A_{0}\left( {\rho^{\ell}e^{i\;{\ell\phi}}} \right)} \cdot p} \right\rbrack}{\psi_{g,k}(r)}{{dr}.}}}}$

The first order Taylor expansion with

=0 is then

$\left( {\hat{M}}_{\rho} \right)_{g,e} = {\left( {\hat{M}}_{\rho} \right)_{g,e}^{0} + {\sum\limits_{\alpha,S}{\frac{h_{es}^{\alpha}Q_{a}}{\Delta\; E_{e,S}}{\left( {\hat{M}}_{\rho} \right)_{g,e}^{0}.}}}}$

Since h_(e,S) ^(α), Q_(a), and ΔE_(es) depend only on the properties ofthe crystal and not

, only M affects scattering intensities when using vortex light.Subsequently, the electronic wavefunction and e are left as relativevalues of M(

≠0) with respect to M(

=0) for the Raman effect with vortex light interactions with crystalsolids.

Raman scattering intensity enhancements may be identified by selectingappropriate values of

such as in the case of zinc blende crystals, for example, in which amaximum was reported for

=30 based on symmetry considerations using the approach presented above.In practice, focusing a laser producing vortex light has little impacton the intensity enhancement of M given its similarity to focusing lightin an ordinary Raman scattering measurement.

Polarized Raman Spectroscopy

Given that the polarizability of molecules varies spatially with respectto the distribution of molecules in a sample, a plane-polarized Ramansource may be used to characterize the atomic structure of crystals andmolecular structure of polymeric films, crystals, and liquid crystals.

Referring now to FIG. 78 , polarized Raman techniques involve apolarizer 7806 between the sample 7804 and the spectrometer 7808oriented either parallel (∥) or perpendicular (⊥) to the polarizationstate of the laser source 7802. As well, polarizing optics 7810 may beinserted between the laser 7802 and sample 7804 to select an appropriatestate of polarization incident on the sample.

The symmetry properties of bond vibrations in a molecule arecharacterized by polarized Raman spectroscopy by evaluating thedepolarization, ρ, of particular intensity peaks,

$\rho = \frac{I_{\bot}}{I_{\parallel}}$where I⁻ ⊥ and I⁻ ∥ are the Raman spectral band intensities withpolarizations perpendicular and parallel, respectively, to the state ofpolarization of the laser source 7802.

As shown in FIG. 79 , information gained by polarized Raman spectroscopy7902 can be used to supplement atomic and molecular information gainedby non-polarized Raman spectroscopy 7904. A single integratedspectroscopy unit 7906 exploiting both polarized and non-polarized Ramaneffects using combined results processing 7908 that improves overallquality and amount of information gained by spectroscopically processingdata from a sample using multiple types of spectroscopic analysis.

Raman Spectroscopy with Optical Vortices

The typical Raman source is a Gaussian laser operating in itsfundamental mode with an electric field

${E\left( {x,y,z} \right)} = {\hat{e}E_{0}\mspace{11mu}{\exp\left( {- \frac{x^{2} + y^{2}}{w^{2}}} \right)}{\exp\left\lbrack {- {i\left( {{kz} - {\omega\; t}} \right)}} \right\rbrack}}$

traveling in the z-direction, where ê is the polarization vector. Lightproduced by such a source has either linear or circular polarizationwhich are limited to the transverse (x, y) plane with no electric fieldcomponent in the z-direction. The induced dipole moments of interestthen are only P_(x) and P_(y).

A longitudinal mode along the z-direction incident on a moleculescatters light that completes the picture of the molecule'spolarizability to include P_(z). An electric field having a z-componentis a radially polarized beam with a polarization vectorê=x{circumflex over (x)}+yŷ={circumflex over (r)}.

Several methods exist to generate radially polarized fields havinglongitudinal components when tightly focused. In Raman spectroscopy, theinduced dipole moment, P_(z), is the result of E_(z) which may increasethe strength of vibrational modes in addition to generating newvibrational modes previously unobserved with conventional Ramanspectroscopy. As shown in FIG. 80 , information gained by Raman beamsendowed with optical vortices 8002 adds a third degree of spectroscopiccapability when coupled with polarized 8004 and non-polarized 8006 Ramanspectroscopy in a combined analysis 8008. Such Raman spectroscopy canalso work in conjunction with incident photons that carry OAM.

THz Spectroscopy

Terahertz spectroscopy is conducted in the far-infrared frequency rangeof the electromagnetic spectrum and is therefore useful for identifyingfar-infrared vibrational modes in molecules. THz spectroscopy canprovide a higher signal-to-noise ratio and wider dynamic range thanfar-infrared spectroscopy due the use of bright light sources andsensitive detectors. This provides for selective detection of weakinter- and intra-molecular vibrational modes commonly occurring inbiological and chemical processes which are not active inIR-spectroscopy. THz spectroscopy may also be used in conjunction withincident photons that carry OAM. Terahertz waves pass through media thatare opaque in the visible and near-IR spectra and are strongly absorbedby aqueous environments.

THz spectroscopy was historically hindered by a lack of appropriatelyhigh-powered light sources. However, access to practical THzspectroscopy in the far-infrared range was permitted by the generationof THz rays based on picosecond and femtosecond laser pulses. Today, THzsources include either short pulse mode (e.g. photoconductive antennas,optical rectifiers) or continuous wave (CW) mode having a wide range ofavailable output power (nanowatts to 10 watts).

Several different types of THz sources are used today to interrogatebiological, chemical and solid-state processes. Sources in the 1-3.5 THzrange are frequently used in biology and medicine, for example, toinvestigate conformational molecular changes. THz spectroscopy is usedtoday as frequently as Raman spectroscopy.

Terahertz Time-Domain Spectroscopy

Terahertz time-domain spectroscopy (THz-TDS) is one of the most widelyused THz techniques which includes coherent emission of single-cycle THzpulses such as provided by a femtosecond laser. The detection of thesepulses occurs at a repetition rate of about 100 MHz.

Two dimensional THz absorption properties of samples are characterizedby a THz imaging technique. This technique was demonstrated in systemsdesigned for THz-TDS based on picosecond pulses as well as systemsutilizing continuous-wave (CW) sources such as a THz-wave parametricoscillator, quantum cascade laser, or optically pumped terahertz laser.THz spectroscopy can be used in conjunction with incident photons thatcarry OAM.

THz pulse imaging provides broad image frequency information between0.1-5 THz while THz CW imaging may be performed in real-time, isfrequency-sensitive, and has a higher dynamic range due to significantlyhigher spectral power density. In both pulse and CW THz imaging thecharacteristics of the light source (coherency, power, and stability)are important. A THz spectrometer may mechanically scan a sample in twodimensions, but the time of each scan scales with sample size. Real timeTHz imaging is often conducted with an array of THz wave detectorscomposed of electro-optic crystals or a pyroelectric camera. Such THzspectroscopy can be used in conjunction with incident photons that carryOAM.

THz imaging suffers from poor resolution as estimated in terms of itsdiffraction limit which is less than a millimeter and from lowtransmission through an aperture resulting in low sensitivity. To exceedthe diffraction limitation near-field microscopy is used to achievesub-wavelength resolution, though low transmission remains an issue.

Fluorescence Spectroscopy

Perturbed by incident light, electrons in molecules at room temperatureare excited from the lowest vibrational energy level 8102 of theelectronic ground state to either the first (S₁) 8104 or second (S₂)8106 vibrational state (FIG. 81 ) and may occupy any one of severalvibrational sub-levels. Each vibrational sub-level has many neighboringrotational energy levels in such close proximity that inter-sub-levelenergy transitions are almost indistinguishable. Consequently, mostmolecular compounds have broad absorption spectra with the exception ofthose having negligible rotational characteristics such as planar andaromatic compounds.

In fluorescence spectroscopy, molecules absorb energy from incidentphotons, obtain a higher vibrational energy sub-level of an excitedstate (S₁ or S₂), then lose their excess vibrational energy throughcollisions and return to the lowest vibrational sub-level of the excitedstate. Most molecules occupying an electronic state above S₂, experienceinternal conversion and decay by collision through the lowestvibrational energy sub-level of the upper state to a higher vibrationalsub-level of a lower excited state having the same energy. The electronscontinue to lose energy until they occupy the lowest vibrational energysub-level of S₁ 8108. The decay of the molecule into any vibrationalenergy sub-level of the ground state causes the emission of fluorescentphotons.

If the absorption and emission process differ from this sequence, thequantum efficiency is less than unity. The “0-0” transition from thelowest vibrational ground state sub-level to the lowest vibrational S₁sub-level 8108 is common to both the absorption and emission phenomenawhile all other absorption transitions occur only with more energy thanany transition in the fluorescence emission. The emission spectrumsubsequently overlaps the absorption spectrum at the incident photonfrequency corresponding to this “0-0” transition while the rest of theemission spectrum will have less energy and equivalently occurs at alower frequency. The “0-0” transition in the absorption and emissionspectra rarely coincide exactly given a small loss of energy due tointeraction of the molecule with surrounding solvent molecules.

Hence, distributions of vibrational sub-levels in S₁ and S₂ are verysimilar since incident photon energy doesn't significantly affect theshape of the molecule. Energy differences between bands in the emissionspectrum will be similar to those in the absorption spectrum andfrequently, the emission spectrum will be approximately a mirror imageof the absorption spectrum. The shape of the emission spectrum is alwaysthe same despite an incident photon frequency shift from that of theincident radiation since the emission of fluorescent photons alwaysoccurs from the lowest vibrational energy sub-level of S₂. If theincident radiation intensity yielding excitation remains constant as thefrequency shifts, the emission spectrum is considered a correctedexcitation spectrum.

The quantum efficiency of most complex molecules is independent of thefrequency of incident photons and the emission is directly correlated tothe molecular extinction coefficient of the compound. In other words,the corrected excitation spectrum of a substance will be the same as itsabsorption spectrum. The intensity of fluorescence emission is directlyproportional to the incident radiation intensity.

Fluorescence spectroscopy results in emission and excitation spectra. Inemission fluoroscopy, the exciting radiation is held at a fixedwavelength and the emitted fluorescent intensity is measured as afunction of emission wavelength. In excitation fluoroscopy, the emissionwavelength is held fixed and the fluorescence intensity is measured as afunction of the excitation wavelength. This type of fluorescencespectroscopy may also be used in conjunction with incident photons thatcarry OAM. Performing both emission and excitation spectra togetheryields a spectral map of the material under interrogation. Materials ofinterest may contain many fluorophores, and different excitationwavelengths are required to interrogate different molecules.

Fluorescence spectrometers analyze the spectral distribution of thelight emitted from a sample (the fluorescence emission spectrum) bymeans of either a continuously variable interference filter or amonochromator. Monochromators used in more sophisticated spectrometersselect the exciting radiation and analyze the sample emission spectra.Such instruments are also capable of measuring the variation of emissionintensity with exciting wavelength (the fluorescence excitationspectrum).

One advantage of fluorescence spectroscopy compared to equivalentabsorption techniques is that the sample may be contained in simple testtubes rather than precision cuvettes without appreciable loss inprecision because of the geometrical configuration of simplefluorimeters in which only the small central region of the cuvette isinterrogated by the detector. Hence, the overall size of the cuvette isless important.

Sensitivity of fluorescence spectroscopy depends largely on theproperties of the measured sample and is typically measured in parts perbillion or trillion for most materials. This remarkable degree ofsensitivity permits reliable detection of very small sample sizes offluorescent materials (e.g. chlorophyll and aromatic hydrocarbons).

Fluorescence spectroscopy is exceptionally specific and less prone tointerference because few materials absorb or emit light (fluoresce) andrarely emit at the same frequency as compounds in the target material.

Fluorescence measurements scale directly with sample concentration overa broad frequency range and can be performed over a range ofconcentrations of up to about one six orders of magnitude without sampledilution or alteration of the sample cell. Additionally, the sensitivityand specificity of fluoroscopy reduces or eliminates the need for costlyand time-consuming sample preparation procedures, thus expediting theanalysis. Overall, fluoroscopy represents a low-cost materialidentification technique owing to its high sensitivity (small samplesize requirement).

Pump-Probe Spectroscopy

Pump-probe spectroscopy is used to study ultrafast phenomena in which apump beam pulse perturbs atomic and molecular constituents of a sampleand a probe beam pulse is used to interrogate the perturbed sample afteran adjustable period of time. This optical technique is a type oftransient spectroscopy in which the electronic and structural propertiesof short-lived transient states of photochemically or photophysicallyrelevant molecules may be investigated. The resulting excited state isexamined by monitoring properties related to the probe beam includingits reflectivity, absorption, luminescence, and Raman scatteringcharacteristics. Electronic and structural changes occurring withinfemto- to pico-second timeframes may be studied using this technique.

Generally, pump-induced states represent higher energy forms of themolecule. These higher energy molecular forms differ from their lowestground state energy states including a redistribution of electronsand/or nuclei.

Within a basic pump probe configuration, a pulse train generated by alaser is split into a pump pulse and a probe pulse using a beamsplitter. The pump pulse interacts with the atoms and molecules in asample. The probe pulse is used to probe the resulting changes withinthe sample after a short period of time between the pulse train and theprobe pulse train. By changing the delay time between pulse trains withan optical delay line, a spectrum of absorption, reflectivity, Ramanscattering, and luminescence of the probe beam may be acquired after thesample to study the changes made by the pump pulse train at detector. Itis possible to obtain information concerning the decay of thepump-induced excitation by monitoring the probe train as a function ofthe relative time delay. The probe train is typically averaged over manypulses and doesn't require a fast photodetector. The temporal resolutionof measurements in pump-probe spectroscopy is limited only by the pulsedurations of each train. In general, the uncertainty in timing must besmaller than the timescale of the structural or electronic processinduced by the pump train.

In two-color pump-probe spectroscopy, the pump and probe beams havedifferent wavelengths produced by two synchronized sources. While thistechnique provides additional capabilities in ultrafast spectroscopy,it's essential to ensure precise source synchronization with a very lowrelative timing jitter.

In comparison with spontaneous Raman scattering intensities, thescattered intensities provided by a pump-probe Raman spectroscopytechnique may be tremendously enhanced with different pump and probefrequencies, Ω and ω. The frequency of the pump beam is changed, whilethe frequency of the probe beam is fixed. The pump beam is used toinduce Raman emission, while the probe beam serves to reveal Ramanmodes. Both the pump and the probe beam traverse a Raman-active mediumin collinearity. When the difference between the pump and probefrequencies coincide with a Raman vibrational mode frequency, ν, of themedium, the weak spontaneous Raman light is amplified by several ordersof magnitude (10-10⁴) due to the pump photon flux. Gain is achieved.

The pump beam is essentially engineered to provide a variety ofperturbative excitations within a wide range of samples. Pump-probespectroscopy is therefore applicable to use within the context of otherspectroscopy techniques including the use of a pump beam endowed withorbital angular momentum as discussed in the next section.

Orbital Angular Momentum (OAM) Spectroscopy

Chiral optics conventionally involved circularly polarized light inwhich a plane polarized state is understood as a superposition ofcircular polarizations with opposite handedness. The right- andleft-handedness of circularly polarized light indicates its spin angularmomentum (SAM), ±h in addition to the polarization one can use thehelicity of the associated electromagnetic field vectors. Itsinteraction with matter is enantiomerically specific. The combinedtechniques would have specific signatures for different materials.

As described more fully herein above, optical vortices occurring inbeams of light introduce helicity in the wavefront surface of theelectromagnetic fields and the associated angular momentum is considered“orbital”. Orbital angular momentum (OAM) of photonic radiation isfrequently called a “twisted” or “helical” property of the beam. Moststudies of OAM-endowed light interactions with matter involve achiralmolecules.

Delocalized OAM within solid materials associated with the envelopewavefunction in a Bloch framework, which may be spatially macroscopic inextent, may be distinguished from local OAM associated with atoms. Thelatter is associated with the Landé g-factor of electronic states andpart of the effective spin while the former is of interest to orbitallycoherent systems (e.g. quantum Hall layers, superconductors, andtopological insulators). Development of these techniques representsopportunities to improve our understanding of scattering and quantumcoherence of chiral electronic states, with potential implications formaterials discovery and quantum information. To this end, theoreticalframeworks describing the OAM-matter interaction, such as withdielectric materials are useful.

OAM-endowed beams of light have been used to induce such delocalizedOAM-states in solids using a time-resolved pump-probe scheme using LGbeams in which the OAM-sensitive dichroism of bulk n-doped (3×10¹⁶ cm⁻³Si) and undoped GaAs (held in a cryostat at 5K) is exploited. Using thismethod, “whirlpools” of electrons were induced and measured with atime-delayed probe beam whose OAM components were detected in a balancedphotodiode bridge. The study demonstrates that time-resolved OAM decayrates (picoseconds to nanoseconds) are doping dependent, differed fromspin and population lifetimes, and longer than anticipated as describedin M. A. Noyan and J. M. Kikkawa, “Time-resolved orbital angularmomentum spectroscopy,” Appl. Phys. Lett. 107 032406 (2015), which isincorporated herein by reference in its entirety.

A simple pump-probe OAM spectroscopy instrument in which the OAM pumpbeam is an

=±1 Laguerre-Gaussian beam cycled between

=+1 and

=−1 at some frequency,

. The pump beam perturbs target molecules in the sample while a directprobe beam is used to interrogate the resulting perturbation. The samplemay be a crystalline solid, amorphous solid, liquid, biological, orinorganic.

The interaction of light exhibiting OAM, an azimuthal photonic flow ofmomentum, with chiral molecules is the subject of several recenttheoretical and experimental reports. On one hand, the strength of theinteraction has been conjectured as negligible, while on the other hand,not only does such an interaction exist, it may be stronger than theinteractions occurring in conventional polarimetry experiments in whichthe direction of linearly polarized light incident on a solution isrotated by some angle characteristic of the solution itself. A fewlimited experimental studies have suggested that the former theoreticalbody of work is correct—that such an interaction is negligible.

Nonetheless, a variety of light-matter interactions involvingOAM-endowed optical beams indicate a broad range of possibilities inspectroscopy including OAM transfer between acoustic and photonic modes,OAM-endowed Raman sideband generation, and the manipulation of colloidalparticles manipulation with optical OAM “tweezers”.

OAM Spectroscopy of Chiral Molecules

Recent experiments using Laguerre-Gaussian (LG) beams of varying integerazimuthal order,

, traveling through a short optical path length of variousconcentrations of glucose, support the theoretical body of worksuggesting the existence of measureable OAM light-matter interactions.These experiments suggest that not only does the interaction exist, butit appears to be stronger than with polarimetry since perturbations ofthe OAM beam occur within a very short optical path length (1-3 cm) thancommonly required in conventional polarimetry studies (>10 cm) to obtaina measurable perturbation of the linear state of polarization.

The Gaussian beam solution to the wave equation and its extension tohigher order laser modes, including Hermite-Gaussian (HG) and commonlystudied in optics labs. Of particular interest, LG modes exhibit spiral,or helical, phase fronts. In addition to spin angular momentum, thepropagation vector includes an orbital angular momentum (OAM) componentoften referred to as vorticity.

A spatial light modulator (SLM) is frequently used to realize hologramsthat modulate the phase front of a Gaussian beam and has renewedinterest in engineered beams for a variety of purposes.

The expression for the electric field of an LG beam in cylindricalcoordinates is

${{{u\left( {r,\theta,z} \right)} = {\left\lbrack \frac{2{pl}}{1 + {\delta_{\sigma,m}{{\pi\left( {\ell + p} \right)}!}}} \right\rbrack^{\frac{1}{2}}\exp{\left\{ {{j\left( {{2p} + \ell + 1} \right)}\left\lbrack {{\psi(z)} - \psi_{0}} \right\rbrack} \right\} \cdot}}}\quad}\frac{\sqrt{2}r}{w^{2}(z)}{L_{p}^{\ell}\left( \frac{2r^{2}}{w^{2}(z)} \right)}{\exp\left\lbrack {{{- {jk}}\frac{r^{2}}{2{q(z)}}} + {i\;{\ell\theta}}} \right\rbrack}$

with w(z) the beam spot size, q(z) a complex beam parameter comprisingevolution of the spherical wavefront and spot size, and integers p and

index the radial and azimuthal modes, respectively. The exp(i

θ) term describes spiral phase fronts. A collimated beam is reflectedoff the SLM appropriately encoded by a phase retarding forked grating,or hologram, like the one shown in FIG. 43 . The generating equation forthe forked hologram may be written as a Fourier series,

${{T\left( {r,\varphi} \right)} = {\sum\limits_{m = {- \infty}}^{\infty}{t_{m}{\exp\left\lbrack {- {{im}\left( {{\frac{2\pi}{D}r\mspace{11mu}\cos\mspace{11mu}\varphi} - {\ell\varphi}} \right)}} \right\rbrack}}}},$where r and φ are coordinates,

is the order of vorticity, and D is the rectilinear grating period farfrom the forked pole. Weights, t_m, of the Fourier components may bewritten in terms of integer-order Bessel functions,t _(m)=(−i)^(m) J _(m)(kβ)exp(ikα).where kα and kβ bias and modulate the grating phase, respectively. Onlya few terms are needed to generate OAM beams, such as −1≤m≤1,

${T\left( {r,\varphi} \right)} = {\frac{1}{2} - {\frac{1}{2}{{\sin\left( {{\frac{2\pi}{D}r\mspace{11mu}\cos\mspace{11mu}\varphi\mspace{11mu}\varphi} - {\ell\varphi}} \right)}.}}}$Molecular Chirality

The chirality of a molecule is a geometric property of its “handedness”characterized by a variety of spatial rotation, inversion, andreflection operations. Conventionally, the degree of chirality ofmolecules was starkly limited to a molecule being either “chiral” or“achiral” in addition to being “left-handed” or “right-handed”. However,this binary scale of chirality doesn't lend well to detailedspectroscopic studies of millions of molecular systems that may bestudied. In its place, a continuous scale of 0 through 100 has beenimplemented for the past two decades called the Continuous ChiralityMeasure (CCM). Essentially, this continuous measure of chiralityinvolves the Continuous Symmetry Measure (CSM) function,

${S^{\prime}(G)} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}{{P_{i} - {\hat{P}}_{i}}}^{2}}}$where G is a particular symmetry group, P_(i) are the points of theoriginal configuration, {circumflex over (P)}_(i) are the correspondingpoints in the nearest G-symmetric configuration, and n is the totalnumber of configuration points.

The objective is to identify a point set, P_(i), having a desiredG-symmetry such that the total normalized displacement from the originalpoint set P_(i) is a minimum. The range of symmetry, 0≤S′(G)≤1, may beexpanded such that S=100S′. The advantages of CCM over other chiralmeasure schemes include its ease of application to a wide variety ofchiral structures including distorted tetrahedra, helicenes, fullerenes,frozen rotamers, knots, and chiral reaction coordinates, as well asbeing a measured without reference to an ideal shape. Unique chiralityvalues are made with reference to nearest symmetry groups (σ or S_(2n)),thus allowing for direct comparison with a wide variety of geometric.

Yet, since the new technique described above discusses the use ofStimulated Raman or Resonant Raman spectroscopy with vector beams (i.e.,beams with “twistedness” plus polarization), the technique can equallybe applied to both chiral and non-chiral molecules.

Raman with Orbital Angular Momentum

The effect of orbital angular momentum on the Raman scattering spectraof glucose has been investigated. Changes have been observed in theRaman spectra, in particular at 2950 cm⁻¹ with L=2 (helical beam) ascompared to L=0 (Gaussian beam). The innovation is that if the sugarmolecules possess some types of chiral symmetry 8208 than there may be adifferential signal 8202 (FIG. 82 ) using OAM 8204 and Raman 8206spectroscopy. The Raman spectra of glucose, sucrose and fructose havealready been collected for the three laser wavelengths 488, 514.5 and632.8 nm from argon-ion and helium neon laser sources, the signals havebeen tabulated and the agreement of each vibration is justified with theother two laser lines. No resonances were observed as would be expectedsince there is no direct electronic absorption with these energies. TheRaman spectra, however, are sensitive to local and global symmetries ofthe molecule at any wavelength. Differential Raman signals will givefundamental information about the interaction of a chiralelectromagnetic field with the sugar molecules, as well as potentiallylead to a selected symmetry resonance for low level glucose detection inthe blood.

The system used for these measurements is a confocal microscope attachedto a 75 cm single stage spectrometer using a grating blazed at 500 nmand 1200 lines/mm groove density. The microscope objective used was 10×magnification. To generate the OAM beam with angular momentum value L=2,a Q plate was incorporated into the system.

Referring now to FIG. 83 there is illustrated the alignment procedure. Alinear polarizer is inserted at step 8302 into the beam path and rotatedat step 8304 until maximum transmission intensity is achieved. A Q-plateis inserted at step 8306 into the beam path and locates at step 8308 thecenter that produces the OAM beam (by observation of the donut). Thecircular polarizer is inserted at step 8310 before the Q-Plate. Thelinear polarizer is placed at step 8312 after the Q-plate to observe the4 lobed structure. Finally, the circular polarizer is rotated at step8314 until the output from final linear polarizer shows donut for allangles of final linear polarization. This procedure is iterative alsoadjusting applied voltage to Q-plate (appx 4 Volts) and the square wavedriving frequency (appx 2 KHz). The measurements are taken without thefinal linear polarizer.

The resulting spectra with L=2 along with a spectra with L=0 (noelements in the beam path) both normalized to the maximum value whichfor both cases is the Raman signal near 2800 cm⁻¹. From thesemeasurements it does show that there are differential intensitiesbetween the two different excitations. At 400 and 550 cm⁻¹ there isalmost a 50 percent increase in scattering intensity while the L=2spectrum shows a few additional shoulders of each of these lines. Mostpronounced is the intensity ratio of the doublet around 2950 cm⁻¹.

The Raman system used for these measurements is alignment restricted.The incorporation of the additional waveplates causes slight walk-offwhich leads to significant collection intensity drop in the confocalsystem. Presumably, normalization would eliminate any alignmentintensity issues, however signal to noise suffers and longerintegrations are required. Long integration times are not alwayspossible or feasible.

These measurements need to be repeated for glucose and also done forfructose. Also needed to be checked is the response to pure circularpolarization without OAM. We should be able to access the alignment andoptimize for the Q-plate operation. Also, to do is use L=1 value andL=20 values of OAM. With promising results, we will use a quarterwaveplate for 488 nm as this laser produces the best spectra in theshortest acquisition times on the system.

Although the higher energy Raman signals are not unique to glucose asthey represent generic carbon and carbon hydrogen bonds present in manyorganic systems, it may prove to be unique to chiral systems.Additionally, the lower energy modes that are more unique to glucose mayshow better differentiation with OAM once the system is better optimizedfor Q-plates.

Raman Detection of Glycated Protein

Hb and Hb-A1c a proteins by Raman spectroscopy using OAM may also beinvestigated. Mammalian blood is considered as connective tissue becauseof its cellular composition and due to its embryonic origin and also dueto the origin and presence of colloidal proteins in its plasma. RedBlood cells and Plasma proteins are the major constituents of blood.These connective tissue components are targets for metabolic stressunder disease conditions and result in the chemical alterations. All theblood components are subjected to excessive metabolic stress underhyperglycemic states. Blood acts a primary transporter of nutrients,gases and wastes. Blood plasma acts as a primary carrier for glucose tothe tissues. Normal pre-prandial plasma glucose levels are 80 mg/dl to130 mg/dl and normal postprandial plasma glucose is <180 mg/dl. TheRenal Threshold for Glucose (RTG) is the physiologic maximum of plasmaglucose beyond which kidneys fail to reabsorb the glucose and getexcreted in urine. This is a condition called glycosuria. Glycosuria isthe key characteristic of Diabetes mellitus (DM). High plasma glucose inDM will cause increased levels of Glycosylated Hemoglobin also known asHbA1c. Under normal physiological conditions HbA1c levels are <7%, thisalso expressed as eAG which should be below 154 mg/dl in Normo-glycemiccondition.

Glycation of Plasma Proteins in DM

Glycation is defined as the non-enzymatic random nonspecific covalentlinking of glucose or other hexose sugar moieties to the proteins. Undernormal blood glucose levels in healthy individuals will have levels <7%Glycated Hemoglobin (HbA1c) in the blood, however under hyperglycemicconditions like DM, its levels will increase. Higher blood glucoselevels can induce glycation of other major proteins of blood plasma likealbumin.

Advantages of Measurement of Glycated Proteins in DM:

Measurements of blood glucose levels only provide the information aboutthe glycemic status of a subject at a given moment, i.e. a diabeticperson with uncontrolled blood sugar levels for several months may yieldnormal blood glucose level if he/she gets the test under fasting stateor with low carbohydrate intake on a given day. However, the measurementof Glycated hemoglobin (HbA1c) levels in blood yield the informationabout average blood sugar levels in patient for past 2 to 3 months.Therefore, it has become a standard clinical practice since past decadeto measure Glycated Hemoglobin in patients with DM with the developmentsensitive and reliable laboratory analyses. We propose the use of Ramanspectroscopic studies on Diabetic blood and its components for thedetection of specific Raman fingerprints that may result fromnon-enzymatic glycosylation of key blood proteins Hemoglobin, plasmaalbumin and others in its native and altered physical states. Theprocess of glycation in proteins induces the chemical alterations,structural modifications, conformational changes. Any or all of thesecan result in special Raman spectral changes which can used as aclinical marker.

Measurements were carried out with a small benchtop OceanOptics Ramansystem with 532 nm excitation.

Raman Spectroscopy of Tryptophan:

The Hemoglobin (tetramer) has 6 residues of Tryptophan thereforeHemoglobin is a fluorescent protein. Tryptophan can undergo glycationand result in conformational changes in Hemoglobin. The tryptophanchanges can be identified by using Raman studies (Masako Na-Gai et al.Biochemistry, 2012, 51 (30), pp 59325941) which is incorporated hereinby reference. In order to understand the glycation induced Ramanspectral changes in Tryptophan residues Raman spectra is obtained fromanalytical grade amorphous Tryptophan using 532 nm OceanOptics Raman.

Raman Spectra of Proteins:

Solid amorphous powders of albumin and Glycated albumin samples weresubjected to Raman measurements using a OceanOptic 532 nm Raman systemand the confocal Raman system using 488, 514.5 and 632.8 nm. No Ramansignal was observed from these samples, and therefore we need to retestin solution at a physiologic pH of 7.4.

The next steps are:

1. NIR Raman: Blood and its components have intense fluorescence invisible range so NIR Raman may help reduce fluorescence and get goodRaman signals from target protein molecules.

2. OceanOptics 532 nm Raman: This can be used detect some of Glycationderivatives in blood. This needs normal and diabetic blood either fromhuman subjects or animal models. And also Reference spectra of syntheticglycation products can be obtained by using this system, which can laterbe compared with the Raman signal from blood samples.

3. In Vivo Animal model: For future experiments to be successful for invivo blood glucose and diabetes testing, the Raman measurements need tobe carried out in a rat diabetes animal model.

OAM with Raman for Food Freshness, Spoilage, and Organic Detection

Another aspect that will be investigated is food safety concerns due tospoilage of meats, produce, diary, and grains and determination iflabeled food is organic using Raman and OAM. Public and individualconcern led to both governmental regulation and commercial requirementsof quality, stability, and safety of food storage periods. Moreover,food deterioration resulting in food spoilage leads to not only healthissues but also economic loss to food manufacturing and relatedindustries. Thus, minimizing food spoilage, determining food freshness,or maximizing shelf life of food is desired.

Moreover, in 2000, the U.S. Department of Agriculture (“USDA”)established guidelines and national standards for the term “organic.”For example, organic food, as defined by USDA guidelines, means thatfood must be produced without sewer-sludge fertilizers, syntheticfertilizers and pesticides, genetic engineering, growth hormones,irradiation, and antibiotics.

The traditional physical characteristics of food spoilage, such asunpleasant smells, unpleasant tastes, color changes, texture changes,and mold growth, manifest well after biochemical processes have occurredthat impair food quality or safety. As a result, they are not adequateindicators of determining acceptable criteria to use for food freshness,preservation, and spoilage.

Thus, research to date includes the identification of so-called“biomarkers” of food spoilage. This research includes identification ofthe biochemical mechanisms that produce certain chemical by-productsthat are associated with the physical characteristics of food spoilage.These mechanisms can be physical (e.g., temperature, pH, light,mechanical damage); chemical (e.g., enzymatic reaction, non-enzymaticreaction, rancidity, chemical interaction); microorganism-based (e.g.,bacteria, viruses, yeasts, molds); or other (e.g., insects, rodents,animals, birds).

One aspect of the investigation is to use OAM and Raman techniques toidentify these so-called biomarkers and their associated concentrationsto better determine shelf life of basic food categories. Additionally,another aspect of the invention is to investigate the chemicals usedthat would fail to qualify foodstuffs as “organic.” For example, theTable 1 below shows several researched biochemical processes andchemical by-products associated with food spoilage mechanisms associatedwith common food groups:

Food Category/ Biochemical Process Mechanism Spoilage Action ResultingBiomarker Oxidation Light Reversion Flavor of 2-pentyl furan SoybeanOxidation Light Sunlight flavor dimethyl disulfide, in milk 2-butanone,ethanol, diacetyl, n-butanol Oxidation Light Loss of Riboflavin, vitaminD-5,6 ep25 Vitamins D, E, and C oxide Oxidation Light Greening of Potatoalpha-solanine, alpha-chaconine Oxidation Decay meat and diary aldehydes(fats, oils, lipids) Enzymatic Decay Chicken/Meat dimethylsulfide,dimethyl disulfide, dimethyl trisulfide, dimethyl tetrasulfide, hydrogensulfide, ethanol, 3-methyl-1-butanol, acetic acid, propanioc acid,methanethiol, free fatty acids (FFAs) Enzymatic: Decay Fruits,Vegatables, biogenic amines Decarboxylation of free Meat, Fish, Poultry(tyraimine, putrescine, amino acids (natural cadaverine, histamine)fermentation or via contimation of microorganisms) Enzymatic DecayVegatables ascrbic acid, oxidase vitamin (loss of C) Enzymatic DecayMilk, oils lipase, glycerol, free (hydrolytic rancidity) fatty acids(FFAs), 3-(E)-hexenal, 2-(E)-hexenal Enzymatic Decay Vegatables (losslipoxygenase of vitamin A) Enzymatic Decay Fruits (loss of pectic peticenzymes substances, i.e., softing) Enzymatic Decay Fruits (browning)peroxidases (polyphenol oxidase, o-diphenol, monophenol, o-quinone)Enzymatic Decay Fruits, Vegatables melanin (browning, sour flavor,vitamin loss) Enzymatic Decay Eggs, Crab, Lobster, proteases Flour(reduction of shelf life, overtenderization, reduction in gluten networkformation) Enzymatic Decay Meats, Fish thiaminase Microbial BacteriaCarbohydrates alcoholic (ethanol, CO2); (fermentation) homofermentativelactic acid (lactic acid); heterofermentative lactic acid (lactic acid,acetic aci, ethanol, CO2); propionic acid fermentation (propionic acid,aetic acid, CO2); butyric acid fermentation (butyric acid, acetic acid,CO2, H2); mixed acid fermentation (lactic acid, acetic acid, CO2, H2,ethanol); 2,3-butanediol fermentation (CO2, ethanol, 2,3- butanediol,formic acid) Microbial Bacteria Degradation of (H2S, methyl mercaptns,N-Compounds indole, cadaverine, putrescine, histamine) Microbialbacteria Fish (odor) trimethylamine Microbial Bacteria Lipids aldehyde,ketones Microbial Bacteria Pectin Degradation polygalcturonic acid,galacturonic acid, methanol Fishy Odor Decay Meat, Egg, Fishtrimethylamine Garlic odor Decay Wine, Fish, Meat, Milk dimethyltrisulfide Onion odor Decay Wine, Fish, Meat, Milk dimethyl disulfideCabbage odor Decay Wine, Fish, Meat, Milk dimethyl sulfide Fruity odorDecay Milk, Fish, Wine esters Potato odor Decay Meat, Egg, Fish2-methoxy-3- isopropylprazine Alcoholic odor Decay Fruit juices, ethanolMayonnaise Musty odor Decay Bread, Wine tricholoranisole Cheesy odorDecay Meat diacetyl, acetoin Medicinal odor Decay Juice, Wine 2-methoxyphenol Souring Decay Wine, Beer, Dairy acetic acid, lactic acid, citricacid Slime Decay Meat, Juices, Wine polysaccharide Curdling Decay Milklactic acid Holes Decay Hard cheese carbon dioxide

A person skilled in the art would be well aware of various othermechanisms and biochemical indicators evidencing food spoilage of commonfoodstuffs, including other reactions or volatile or non-volatileorganic compound (VOC) by-products associated with food spoilage.Likewise, a person skilled in the art would be well aware of thechemicals and additives that do not qualify food as organic, whetherinvestigating grains, diary, produce, or meats.

Traditional spectroscopy techniques are not adequate to identify inreal-time or adequate concentration these biomarkers in any meaningfulmanner to determine shelf life of the food sample or organic nature ofthe food in question. The present investigation and invention willemploy Raman and OAM techniques described above to classify, identify,and quantify the various bio-markers in the table above and the commonchemicals that do not qualify food as organic as defined in federalregulations.

Such techniques are equally applicable whether the biomarker or chemicalis a chiral or non-chiral molecule. Such data can then be correlated toconcentration of degradation of the sampled food group to determineminimum and maximum concentrations acceptable to food freshness,spoilage, organic quality, and safety.

Ince-Gaussian Spectroscopy

Another type of spectroscopic technique that may be combined with one ormore other spectroscopic techniques is Ince-Gaussian Spectroscopy. InceGaussian (IG) beams are the solutions of paraxial beams in an ellipticalcoordinate system. IG beams are the third calls of orthogonal Eigenstates and can probe the chirality structures of samples. Since IG modeshave a preferred symmetry (long axis versus short axis) this enables itto probe chirality better than Laguerre Gaussian or Hermite Gaussianmodes. This enables the propagation of more IG modes than LaguerreGaussian modes or Hermite Gaussian modes. Thus, IG modes can be used asa program signal for spectroscopy in the same manner that LaguerreGaussian modes or Hermite Gaussian modes are used. This enables thedetection of types of materials and concentration of materials using anIG mode probe signal.

The wave equation can be represented as a Helmholtz equation inCartesian coordinates as follows(∇² +k ²)E(x,y,z)=0E(x, y, z) is complex field amplitude which can be expressed in terms ofits slowly varying envelope and fast varying part in z-direction.E(x,y,z)=ψ(x,y,z)e ^(jkz)

A Paraxial Wave approximation may be determined by substituting ourassumption in the Helmholtz Equation.

${\left( {\nabla^{2}{+ k^{2}}} \right){\psi \cdot e^{jkz}}} = {{{0\frac{\delta^{2}\psi}{\delta\; x^{2}}} + \frac{\delta^{2}\psi}{\delta\; y^{2}} + \frac{\delta^{2}\psi}{\delta\; z^{2}} - {j\; 2k\frac{\delta\psi}{\delta\; z}}} = 0}$

We then make our slowly varying envelope approximation

${{\frac{\delta^{2}\psi}{\delta\; z^{2}}} ⪡ {\frac{\delta^{2}\psi}{\delta\; x^{2}}}},{\frac{\delta^{2}\psi}{\delta\; y^{2}}},{2k{\frac{\delta\psi}{\delta\; z}}}$${{\nabla_{t}^{2}\psi} + {j\; 2\; k\frac{\delta\psi}{\delta\; z}}} = 0$

Which comprises a Paraxial wave equation.

The elliptical-cylindrical coordinate system may be defined:

x = a cosh  ξcosη y = a sinh  ξsinη ξ ∈ (0, ∞), η ∈ (0, 2π)$a = {{{f(z)}\mspace{14mu}{where}\mspace{14mu}{f(z)}} = \frac{f_{0}{w(z)}}{w_{0}}}$

Curves of constant value of ξ trace confocal ellipses.

${{\frac{x^{2}}{a^{2}\cosh^{2}\xi} + \frac{y^{2}}{a^{2}\sinh^{2}\xi}} = 1}\mspace{14mu}({Ellipse})$

A constant value of η give confocal hyperbolas as shown in FIG. 84 .

${{\frac{x^{2}}{a^{2}\cos^{2}\eta} - \frac{y^{2}}{a^{2}\sin^{2}\eta}} = 1}\mspace{14mu}({hyperbola})$

An elliptical-cylindrical coordinate system may then be defined in thefollowing manner

$\nabla_{t}^{2}{= {{\frac{1}{h_{\xi}^{2}}\frac{\delta^{2}}{{\delta\xi}^{2}}} + {\frac{1}{h_{\eta}^{2}}\frac{\delta^{2}}{{\delta\eta}^{2}}}}}$

Where h_(ξ), h_(η) are scale factors

$h_{\xi} = \sqrt{\left( \frac{\delta\; x}{\delta\;\xi} \right)^{2} + \left( \frac{\delta\; y}{\delta\;\xi} \right)^{2}}$$h_{\eta} = \sqrt{\left( \frac{\delta\; x}{\delta\;\eta} \right)^{2} + \left( \frac{\delta\; y}{\delta\;\eta} \right)^{2}}$$h_{\xi} = {h_{\eta} = {a\sqrt{{\sinh^{2}\xi} + {\sin^{2}\eta}}{\nabla_{t}^{2}{= {\frac{1}{a^{2}\sinh^{2}{\xi sin}^{2}\eta}\left( {\frac{\delta^{2}}{{\delta\xi}^{2}} + \frac{\delta^{2}}{\delta\eta^{2}}} \right)}}}}}$

The solution to the paraxial wave equations may then be made inelliptical coordinates. Paraxial Wave Equation in Elliptic Cylindricalco-ordinates are defined as

${{\frac{1}{a^{2}\left( {\sinh^{2}{\xi sin}^{2}\eta} \right)}\left( {\frac{\delta^{2}\psi}{{\delta\xi}^{2}} + \frac{\delta^{2}\psi}{\delta\eta^{2}}} \right)} - {j2k\frac{\delta\psi}{\delta z}}} = 0$

Assuming separable solution as modulated version of fundamental Gaussianbeam.IG(

)=E(ξ)N(η)exp(jZ(z))ψ_(GB)(

)

Where

${\psi_{GB}\left( r^{\sim} \right)} = {\frac{w_{0}}{w(z)}{\exp\left\lbrack {{- \frac{r^{2}}{w^{2}(z)}} + {j\frac{{kr}^{2}}{2{R(z)}}} - {j{\psi_{GS}(z)}}} \right\rbrack}}$E, N & Z are real functions. They have the same wave-fronts as ψ_(GB)but different intensity distribution.

Separated differential equations are defined as

${\frac{d^{2}E}{d\;\xi^{2}} - {{\epsilon sinh2\xi}\frac{dE}{d\xi}} - {\left( {a - {p\;\epsilon\;\cosh\; 2\xi}} \right)E}} = 0$${\frac{d^{2}N}{d\eta^{2}} - {\epsilon\;\sin\; 2\eta\frac{dN}{d\eta}} - \left( {a - {p\;{\epsilon cos2\eta}}} \right)} = {{0 - {\left( \frac{z^{2} + z_{\gamma}^{2}}{z_{r}} \right)\frac{dZ}{dz}}} = p}$

Where a and p are separation constants

$\epsilon = \frac{f_{0}w_{0}}{w(z)}$

The even solutions for the Ince-Gaussian equations are

${{IG}_{pm}^{e}\left( {r^{\sim},\epsilon} \right)} = {\frac{Cw_{o}}{w(z)}{C_{p}^{m}\left( {{j\;\xi},\epsilon} \right)}{C_{p}^{m}\left( {\eta,\epsilon} \right)}{\exp\left( {- \frac{r^{2}}{w^{2}(z)}} \right)} \times \exp{j\left( {{kz} + \frac{kr^{2}}{2{R(z)}} - {\left( {p + 1} \right){\psi_{GS}(z)}}} \right)}}$

The odd solutions for the Ince-Gaussian equations are

${{IG}_{pm}^{o}\left( {r^{\sim},\epsilon} \right)} = {\frac{sw_{0}}{w(z)}{S_{p}^{m}\left( {{j\;\xi},\epsilon} \right)}{S_{p}^{m}\left( {\eta,\epsilon} \right)}{\exp\left( {- \frac{r^{2}}{w^{2}(z)}} \right)} \times \exp\;{j\left( {{kz} + \frac{kr^{2}}{2{R(z)}} - {\left( {p + 1} \right){\psi_{GS}(z)}}} \right)}}$

Thus, as previously discussed, by combining two or more different typesof spectroscopy techniques, various types of different parameters may bemonitored and used for determining types and concentrations of samplematerials. The use of multiple types of spectroscopic parameter analysisenables for more accurate and detailed analysis of sample types andconcentrations. Thus, any number of spectroscopic techniques such asoptical spectroscopy, infrared spectroscopy, Ramen spectroscopy,spontaneous Ramen spectroscopy, simulated Ramen spectroscopy, resonanceRamen spectroscopy, polarized Ramen spectroscopy, Ramen spectroscopywith optical vortices, THz spectroscopy, terahertz time domainspectroscopy, fluorescence spectroscopy, pump probe spectroscopy, OAMspectroscopy, or Ince Gaussian spectroscopy may be used in any number ofvarious combinations in order to provide better detection of sampletypes in concentrations. It should be realized that the types ofspectroscopy discussed herein are not limiting in any combination ofspectroscopic techniques may be utilized in the analysis of samplematerials.

Using the orbital angular momentum state of the transmitted energysignals, physical information can be embedded within the electromagneticradiation transmitted by the signals. The Maxwell-Heaviside equationscan be represented as:

${{\nabla E} = \frac{\rho}{ɛ_{0}}}{{\nabla \times E} = {- \frac{\partial B}{\partial t}}}$∇⋅B = 0${\nabla{\times B}} = {{ɛ_{0}\mu_{0}\frac{\partial E}{\partial t}} + {\mu_{0}{j\left( {t,x} \right)}}}$thewhere ∇ is the del operator, E is the electric field intensity and B isthe magnetic flux density. Using these equations, we can derive 23symmetries/conserve quantities from Maxwell's original equations.However, there are only ten well-known conserve quantities and only afew of these are commercially used. Historically if Maxwell's equationswhere kept in their original quaternion forms, it would have been easierto see the symmetries/conserved quantities, but when they were modifiedto their present vectorial form by Heaviside, it became more difficultto see such inherent symmetries in Maxwell's equations.

The conserved quantities and the electromagnetic field can berepresented according to the conservation of system energy and theconservation of system linear momentum. Time symmetry, i.e. theconservation of system energy can be represented using Poynting'stheorem according to the equations:

$H = {{\sum{m_{i}\gamma_{i}c^{2}}} + {\frac{ɛ_{0}}{2}{\int{d^{3}{x\left( {{E}^{2} + {c^{2}{B}^{2}}} \right)}}}}}$${\frac{dU^{mech}}{dt} + \frac{dU^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}{\hat{n^{\prime}} \cdot S}}}} = 0$

The space symmetry, i.e., the conservation of system linear momentumrepresenting the electromagnetic Doppler shift can be represented by theequations:

$P = {{\sum\limits_{i}{m_{i}\gamma_{i}v_{i}}} + {ɛ_{0}{\int{d^{3}{x\left( {E \times B} \right)}}}}}$${\frac{dp^{mech}}{dt} + \frac{dp^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}\hat{n^{\prime} \cdot}T}}} = 0$

The conservation of system center of energy is represented by theequation:

$R = {{\frac{1}{H}{\sum_{i}{\left( {x_{i} - x_{0}} \right)m_{i}\gamma_{i}c^{2}}}} + {\frac{ɛ_{0}}{2H}{\int{d^{3}{x\left( {x - x_{0}} \right)}\left( {{E^{2}} + {c^{2}{B^{2}}}} \right)}}}}$Similarly, the conservation of system angular momentum, which gives riseto the azimuthal Doppler shift is represented by the equation:

${\frac{{dJ}^{mech}}{dt} + \frac{{dJ}^{em}}{dt} + {\oint_{s^{\prime}}{d^{2}x^{\prime}n^{\hat{\prime}}}}}{{\cdot M} = 0}$

For radiation beams in free space, the EM field angular momentum J^(em)can be separated into two parts:

J^(em) = ɛ₀∫_(V^(′))d³x^(′)(E × A) + ɛ₀∫_(V^(′))d³x^(′)E_(i)[(x^(′) − x₀) × ∇]A_(i)

For each singular Fourier mode in real valued representation:

$J^{em} = {{{- i}\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}{x^{\prime}\left( {E^{*} \times E} \right)}}}} - {i\frac{ɛ_{0}}{2\omega}{\int_{V^{\prime}}{d^{3}x^{\prime}{E_{i}\left\lbrack {\left( {x^{\prime} - x_{0}} \right) \times \nabla} \right\rbrack}E_{i}}}}}$

The first part is the EM spin angular momentum S^(em), its classicalmanifestation is wave polarization. And the second part is the EMorbital angular momentum L^(em) (or other orthogonal function) itsclassical manifestation is wave helicity. In general, both EM linearmomentum P^(em), and EM angular momentum J^(em)=L^(em)+S^(em) areradiated all the way to the far field.

By using Poynting theorem, the optical vorticity of the signals may bedetermined according to the optical velocity equation:

${{\frac{\partial U}{\partial t} + {\nabla{\cdot S}}} = 0},$where S is the Poynting vectorS=¼(E×H*+E*×H),and U is the energy densityU=¼(ε|E| ²+μ₀ |H| ²),with E and H comprising the electric field and the magnetic field,respectively, and ε and μ₀ being the permittivity and the permeabilityof the medium, respectively. The optical vorticity V may then bedetermined by the curl of the optical velocity according to theequation:

$V = {{\nabla{\times v_{opt}}} = {\nabla{\times \left( \frac{{E \times H^{*}} + {E^{*} \times H}}{{ɛ{E}^{2}} + {\mu_{0}{H}^{2}}} \right)}}}$

The creation and detection of one embodiment using orbital angularmomentum signals comprises one embodiment. However, any orthogonalfunction can be used, including Jacobi functions, Gegenbauer functions,Legendre functions, Chebyshev functions, Laguerre functions, Gaussianfunctions, ect.

These techniques are more fully described in U.S. Pat. No. 9,662,019,entitled Orbital Angular Momentum and Fluorescence-Based MicroendoscopeSpectroscopy for Cancer Diagnosis, filed on Apr. 8, 2015, which isincorporated herein by reference.

One manner for applying the resonance signal from the device to thesurface or area to be sanitized involves the use of patch antenna arraysin the microwave band and within the photonic bands using DLPtechnologies or spiral phase plates. The patch antenna arrays could bethose discussed in corresponding U.S. Pat. No. 10,608,768, entitledPatch Antenna Array for Transmission of Hermite-Gaussian and LaguerreGaussian Beams, filed on Jul. 17, 2018, which is incorporated herein byreference.

With respect to the patch antenna arrays, FIGS. 84 and 85 illustrates amultilayer patch antenna array 8402. The multilayer patch antenna array8402 includes a first antenna layer 8404 for transmitting a firstordered beam, a second antenna layer 8406 for transmitting a secondordered beam and a third layer 8408 for transmitting a third orderedbeam. Each of the layers 8404, 8406 and 8408 are stacked on a samecenter. While the present embodiment is illustrated with respect to amultilayer patch antenna array 8402 including only three layers, itshould be realized that either more or less layers may be implemented ina similar fashion as described herein. On the surface of each of thelayers 8404, 8406 and 8408 are placed patch antennas 8410. Each of thepatch antennas are placed such that they are not obscured by the abovelayer. The layers 8404, 8406 and 8408 are separated from each other bylayer separator members 8412 that provide spacing between each of thelayers 8404, 8406 and 8408. The configuration of the layers of the patchantenna may be in rectangular, circular or elliptical configurations togenerate Hermite-Gaussian, Laguerre-Gaussian or Ince-Gaussian beams.

The patch antennas 8410 used within the multilayer patch antenna array8402 are made from FR408 (flame retardant 408) laminate that ismanufactured by Isola Global, of Chandler Ariz. and has a relativepermittivity of approximately 3.75. The antenna has an overall height of125 μm. The metal of the antenna is copper having a thickness ofapproximately 12 μm. The patch antenna is designed to have an operatingfrequency of 73 GHz and a free space wavelength of 4.1 mm. Thedimensions of the input 50 Ohm line of the antenna is 280 μm while theinput dimensions of the 100 Ohm line are 66 μm.

Each of the patch antennas 8410 are configured to transmit signals at apredetermined phase that is different from the phase of each of theother patch antenna 8410 on a same layer. Thus, as further illustratedin FIG. 86 , there are four patch antenna elements 8410 included on alayer 8404. Each of the antenna elements 8404 have a separate phaseassociated there with as indicated in FIG. 86 . These phases includeπ/2, 2(π/2), 3(π/2) and 4(π/2). Similarly, as illustrated in FIG. 4layer 8406 includes eight different patch antenna elements 8410including the phases π/2, 2(π/2), 3(π/2), 4(π/2), 5(π/2), 6(π/2), 7(π/2)and 8(π/2) as indicated. Finally, referring back to FIG. 84 , there areincluded 12 patch antenna elements 8410 on layer 8408. Each of thesepatch antenna elements 8410 have a phase assigned thereto in the mannerindicated in FIG. 84 . These phases include π/2, 2(π/2), 3(π/2), 4(π/2),5(π/2), 6(π/2), 7(π/2), 8(π/2), 9(π/2), 10(π/2), 11(π/2) and 12(π/2).

Each of the antenna layers 8404, 8406 and 8408 are connected to acoaxial end-launch connector 8416 to feed each layer of the multilayerpatch antenna array 8402. Each of the connectors 8416 are connected toreceive a separate signal that allows the transmission of a separateordered antenna beam in a manner similar to that illustrated in FIG. 85. The emitted beams are multiplexed together by the multilayered patchantenna array 8402. The orthogonal wavefronts transmitted from eachlayer of the multilayered patch antenna array 8402 in a spatial mannerto increase capacity as each wavefront will act as an independent Eigenchannel. The signals are multiplexed onto a single frequency andpropagate without interference or crosstalk between the multiplexedsignals. While the illustration with respect to FIG. 85 illustrates thetransmission of OAM beams at OAM 1, OAM 2 and OAM 3 ordered levels.

It should be understood that other types of Hermite Gaussian andLaguerre Gaussian beams can be transmitted using the multilayer patchantenna array 8402 illustrated. Hermite-Gaussian polynomials andLaguerre-Gaussian polynomials are examples of classical orthogonalpolynomial sequences, which are the Eigenstates of a quantum harmonicoscillator. However, it should be understood that other signals may alsobe used, for example orthogonal polynomials or functions such as Jacobipolynomials, Gegenbauer polynomials, Legendre polynomials and Chebyshevpolynomials. Legendre functions, Bessel functions, prolate spheroidalfunctions and Ince-Gaussian functions may also be used. Q-functions areanother class of functions that can be employed as a basis fororthogonal functions.

The feeding network 8418 illustrated on each of the layers 8404, 8406,8408 uses delay lines of differing lengths in order to establish thephase of each patch antenna element 8410. By configuring the phases asillustrated in FIG. 87 the OAM beams of different orders are generatedand multiplexed together.

Referring now to FIG. 88 , there is illustrated a transmitter 8802 forgenerating a multiplexed beam for transmission toward a virus. Asdiscussed previously, the multilayered patch antenna array 8402 includesa connector 8416 associated with each layer 8404, 8406, 8408 of themultilayer patch antenna array 8402. Each of these connectors 8416 areconnected with signal generation circuitry 8804. The signal generationcircuitry 8804 includes, in one embodiment, a 60 GHz local oscillator8806 for generating a 60 GHz carrier signal. The signal generationcircuit 8804 may also work with other frequencies, such as 84/80 GHz.The 60 GHz signal is output from the local oscillator 8806 to a powerdivider circuit 8808 which separates the 60 GHz signal into threeseparate transmission signals. Each of these separated transmissionsignals are provided to an IQ mixer 8810 that are each connected to oneof the layer input connectors 8416. The IQ mixer circuits 8810 areconnected to an associated additive white gaussian noise circuit 8812for inserting a noise element into the generated transmission signal.The AWG circuit 8812 may also generate SuperQAM signals for insertioninto the transmission signals. The IQ mixer 8810 generates signals in amanner such as that described in U.S. patent application Ser. No.14/323,082, filed on Jul. 3, 2014, now U.S. Pat. No. 9,331,875, issuedon May 3, 2016, entitled SYSTEM AND METHOD FOR COMMUNICATION USINGORBITAL ANGULAR MOMENTUM WITH MULTIPLE LAYER OVERLAY MODULATION, whichis incorporated herein by reference in its entirety.

Using the transmitter 8802 illustrated in FIG. 88 . A multiplexed beam(Hermite Gaussian, Laguerre Gaussian, etc.) can be generated. Themultilayered patch antenna array 8402 will generate a multiplexed beamfor transmission. In the present example, there are multiplex OAM beamthat has twists for various order OAM signals in a manner similar tothat disclosed in U.S. patent application Ser. No. 14/323,082, which isincorporated herein by reference. An associated receiver detector woulddetect the various OAM rings wherein each of the rings is associatedwith a separate OAM processed signal.

When signals are transmitted in free space (vacuum), the signals aretransmitted as plane waves. They may be represented as described hereinbelow. Free space comprises a nonconducting medium (σ=0) and thusJ=σE=0.

From experimental results Ampere's law and Faraday's law are representedas:

${{{{{{\overset{\rightarrow}{B} = {\mu\;\overset{\rightarrow}{H}}}{{\nabla{\times H}} = {\frac{\partial D}{\partial t} + {J\mspace{14mu}{Ampere}}}}}’}s}{\overset{\rightarrow}{D} = {{\epsilon\;\overset{\rightarrow}{E}\overset{\rightarrow}{J}} = {\sigma\;\overset{\rightarrow}{E}}}}{{\nabla{\times E}} = {\frac{- {\partial B}}{\partial t}\mspace{14mu}{Faraday}}}}’}s$If there is propagation in the z direction and therefore E and H are inthe xy plane.

Without the loss of any generality E may be oriented in the x-directionand H may be oriented in the y-direction thus providing propogation inthe z-direction. From Ampere's-Maxwell equation, the following equationsare provided:

${\nabla{\times H}} = \frac{\partial D}{\partial t}$${\nabla{\times H}} = {\begin{matrix}\hat{x} & \hat{y} & \hat{z} \\\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\H_{x} & H_{y} & H_{z}\end{matrix}}$${\left( {\frac{\partial{Hz}}{\partial y} - \frac{\partial{Hy}}{\partial z}} \right)^{\hat{x}} + \left( {\frac{\partial{Hz}}{\partial z} - \frac{\partial{Hz}}{\partial z}} \right)^{\hat{y}} + \left( {\frac{\partial{Hy}}{\partial x} - \frac{\partial{Hx}}{\partial y}} \right)^{\hat{z}}} = {\frac{\partial}{\partial t}\epsilon\; E}$

Next, the vectorial wave equations may be represented as:

$\mspace{20mu}{{\nabla{\times H}} = {\frac{\partial D}{\partial t} + J}}$$\mspace{20mu}{{\nabla{\times H}} = {\epsilon\frac{\partial E}{\partial t}}}$$\mspace{20mu}{{\nabla{\times E}} = \frac{- {\partial B}}{\partial t}}$$\mspace{20mu}{{\nabla{\times E}} = {{- \mu}\frac{\partial H}{\partial t}}}$  ∇×B = 0   ∇×E = S   ∇×∇×H = ∇(∇H) − ∇²H = −∇²H  ∇×∇×E = ∇(∇E) − ∇²E = −∇²E$\mspace{20mu}{{\nabla{\times \left( {\nabla{\times H}} \right)}} = {{\nabla{\times \left( {\epsilon\frac{\partial E}{\partial t}} \right)}} = {{\epsilon\frac{\partial}{\partial t}\left( {\nabla{\times E}} \right)} = {{- {\epsilon\mu}}\frac{\partial}{\partial t}\left( {\frac{\partial}{\partial t}H} \right)}}}}$$\mspace{20mu}{{\nabla^{2}H} = {{+ {\epsilon\mu}}\frac{\partial^{2}}{\partial t^{2}}H}}$$\mspace{20mu}{{{\nabla^{2}H} - {{\epsilon\mu}\frac{\partial^{2}}{\partial t^{2}}H}} = {{0{\nabla{\times \left( {\nabla{\times E}} \right)}}} = {{\nabla{\times \left( {{- \mu}\frac{\partial}{\partial t}H} \right)}} = {{{- \mu}\frac{\partial}{\partial t}\left( {\nabla{\times H}} \right)} = {{{{- \mu}\frac{\partial}{\partial t}\left( {\epsilon\frac{\partial E}{\partial t}} \right)} + {\nabla^{2}E}} = {{+ {\mu\epsilon}}\frac{\partial^{2}}{\partial t^{2}}E}}}}}}$$\mspace{20mu}{{{\nabla^{2}E} - {{\mu\epsilon}\frac{\partial^{2}}{\partial t^{2}}E}} = 0}$

Therefore, in general:{right arrow over (∇)}{right arrow over (E)}+{right arrow over (K)} ²{right arrow over (E)}=0 E({right arrow over (r)},t){right arrow over (E)}(r,t)={right arrow over (E)}({right arrow over(r)})e ^(−jwt) e ^(jkz) Propagating in z-directionTherefore:

${{\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right){\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}e^{- {jwt}}e^{jkz}} + {\frac{W^{2}}{y^{2}}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}e^{- {jwt}}e^{jkz}}} = 0$In free space

$w = {\frac{1}{\sqrt{\mu\epsilon}} = {\left. \rightarrow c \right. = \frac{1}{\sqrt{{\mu\epsilon}\; o}}}}$$k^{2} = \frac{w^{2}}{c^{2}}$Now:

$\mspace{20mu}{{\frac{\partial}{\partial z}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}e^{jkz}} = {e^{jkz}\left\lbrack {\frac{\partial{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}{\partial z} + {{jk}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}} \right\rbrack}}$${\frac{\partial}{\partial z^{2}}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}e^{jkz}} = {{{e^{jkz}\left\lbrack {\frac{\partial{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}{\partial z} + {{jk}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}} \right\rbrack} + {e^{jkz}\left\lbrack {\frac{\partial^{2}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}{\partial z^{2}} + {{jk}\frac{\partial{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}{\partial z}}} \right\rbrack}} = {{e^{jkz}\left\lbrack {{{jk}\frac{\partial\overset{\rightarrow}{E}}{\partial z}} - {k^{2}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}} \right\rbrack} + {e^{jkz}\left\lbrack {\frac{\partial^{2}\overset{\rightarrow}{E}}{\partial z^{2}} + {{jk}\frac{\partial\overset{\rightarrow}{E}}{\partial z}}} \right\rbrack}}}$Because

${{2k\frac{\partial E}{\partial z}}} ⪢ {\frac{\partial^{2}{E(r)}}{\partial z^{2}}}$Paraxial assumption

$\frac{{\partial^{2}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}e^{jkz}}{\partial z^{2}} = {e^{jkz}\left\lbrack {{2{jk}\frac{\partial^{2}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}{\partial z}} - {k^{2}{\overset{\rightarrow}{E}\left( \overset{\rightarrow}{r} \right)}}} \right\rbrack}$Then:

${\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + {2{jk}\frac{\partial^{2}}{\partial z}}} \right){E\left( {x,y,z} \right)}} = 0$Which may be represented in cylindrical coordinates as:

${\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} = {{\frac{1}{q}\frac{\partial}{\partial q}\left( {q\frac{\partial}{\partial q}} \right)} + {\frac{1}{q^{2}}\frac{\partial^{2}}{\partial\Phi^{2}}}}$

This provides a paraxial wave equation in cylindrical coordinates:

$\begin{matrix}{{{\frac{1}{q}\frac{\partial}{\partial q}\left( {q\frac{\partial}{\partial q}} \right){E\left( {q,\Phi,z} \right)}} + {\frac{1}{q^{2}}\frac{\partial^{2}}{\partial\Phi^{2}}{E\left( {q,\Phi,z} \right)}} + {2{jk}\frac{\partial E}{\partial z}\left( {q,\Phi,z} \right)}} = o} \\{{P(z)},{q(z)}}\end{matrix}$Then:

$E_{0}\text{\textasciitilde}e^{- {j{\lbrack{p + {\frac{k}{2q}{({x^{2} + y^{2}})}}}\rbrack}}}$

In general, E_(o) can rotate on the xy-plane and the wave stillpropagates in the z-direction.

$\begin{matrix}{\frac{\partial q}{\partial z} = 1} \\{\frac{\partial P}{\partial z} = {- \frac{j}{q}}}\end{matrix}$q˜Curvature of the phase front near the optical axis.q ₂ =q ₁ +zwhere q₂ is the output plane and q₁ is the input plane. ∞∞

$\frac{1}{q} = {\frac{1}{R} - {j\frac{\lambda}{\pi\; W^{2}}}}$where

$\frac{1}{R}$is the curvature of the wavefront intersecting the z-axis.

Thus, for a complete plane wave R=∞, the equation becomes:

$\frac{1}{q} = {\frac{1}{\left. R\rightarrow\infty \right.} - {j\frac{\lambda}{{\pi W}^{2}}}}$$q_{0} = {\frac{{\pi W}^{2}}{{- j}\;\lambda} = \frac{j\;{\pi W}^{2}}{\lambda}}$where W_(o) is the beam waist.

$\begin{matrix}{q = {{q_{0} + z} = {\frac{j\;{\pi W}_{0}^{2}}{\lambda} + z}}} \\{{w(z)} = {w_{0}\sqrt{1 + \left( \frac{z}{z_{r}} \right)^{2}}}} \\{{W^{2}(z)} = {W_{0}^{2}\left\lbrack {1 + \left( \frac{\lambda z}{{\pi W}_{0}^{2}} \right)^{2}} \right\rbrack}} \\{{R(z)} = {z\left\lbrack {1 + \left( \frac{{\pi W}_{0}^{2}}{\lambda\; z} \right)^{2}} \right\rbrack}} \\{{R(z)} = {z\left\lbrack {1 + \left( \frac{z^{R}}{z} \right)^{2}} \right\rbrack}} \\{{\Phi(z)} = {\tan^{- 1}\left( \frac{z}{z_{R}} \right)}} \\{\theta = \frac{\lambda}{{\pi w}_{0}}} \\{z = z_{R}} \\{{w(z)} = {\sqrt{2}w_{0}}}\end{matrix}$

The Rayleigh length is.

$z_{R} = \frac{\pi\; n}{\lambda_{0}}$where n is the index of refraction.

$\begin{matrix}{w_{0}^{2} = \frac{w^{2}}{1 + \left( \frac{\pi\; w^{2}}{\lambda\; R} \right)^{2}}} \\{z = \frac{R}{1 + \left( \frac{\lambda\; R}{\pi\; w^{2}} \right)^{2}}}\end{matrix}$

The complex phase shift is represented by:

${{{jP}(z)} - {{Ln}\left\lbrack {1 - {j\left( \frac{\lambda\; z}{{\pi w}_{0}^{2}} \right)}} \right\rbrack}} = {{{Ln}\sqrt{1 + \left( \frac{\lambda\; z}{\pi\; w_{0}^{2}} \right)^{2}}} - {j\mspace{14mu}\tan^{- 1}\frac{\lambda\; z}{\pi\; w_{0}^{2}}}}$

The real part of P(z) represents a phase shift difference between theGaussian beam and an ideal plane wave. Thus, the fundamental mode isprovided:

$\begin{matrix}{{E_{0}\left( {x,y,z} \right)} = {{E_{0}\left( {r,z} \right)}\frac{w_{0}}{w}e^{- {j{({{jz} - \phi})}}}e^{- {r^{2}{({\frac{1}{w^{2}} + \frac{jk}{2R}})}}}}} \\{{where}\text{:}} \\{\phi = {\tan^{- 1}\frac{\lambda\; z}{\pi\; w_{0}^{2}}}}\end{matrix}$

Higher order modes may also provide other solutions. The solution ofrectangular equation:

${\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + {2{jk}\frac{\partial}{\partial z}}} \right){E\left( {x,y,z} \right)}} = 0$Can be determined in rectangular coordinates to be:

$\begin{matrix}{{E\left( {x,y,z} \right)} = {\quad{\quad{\underset{mn}{{\quad\quad}\sum}\; C_{nm}E_{0}\frac{w_{0}}{w(z)}{H_{m}\left\lbrack \frac{\sqrt{2}x}{w(z)} \right\rbrack}{H_{n}\left\lbrack \frac{\sqrt{2}y}{w(z)} \right\rbrack}e^{\frac{- {({x^{2} + y^{2}})}}{{w{(t)}}^{2}}}e^{{- {j{({m + m + 1})}}}\tan^{{- 1}\frac{z}{z_{0}}}}e^{j\frac{k\;{({x^{2} + y^{2}})}}{2{R{(z)}}}}}}}} \\{z_{0} = \frac{{kw}_{0}^{2}}{2}} \\{{w\text{(z)}} = {w_{0}\sqrt{1 + \frac{z^{2}}{z_{0}^{2}}}}} \\\left. C_{60}\Rightarrow{TEM}_{OD} \right. \\{{R(z)} = {{z + \frac{z_{0}^{2}}{z}} = {{\frac{z_{0}^{2}}{z}\left( {1 + \frac{z^{2}}{z_{0}^{2}}} \right)} = {{\frac{z_{0}^{2}}{{zw}_{0}^{2}}{w^{2}(z)}} = {\frac{{kz}_{0}}{2z}{w^{2}(z)}}}}}}\end{matrix}$

The solution of cylindrical coordinates of equation:

${{\frac{1}{\rho}\frac{\partial}{\partial\rho}\left( {\rho\frac{\partial}{\partial\rho}} \right){E\left( {\rho,\varnothing,z} \right)}} + {\frac{1}{\rho^{2}}\frac{{\partial{\hat{}2}}{E\left( {\rho,\varnothing,z} \right)}}{{\delta\varnothing}^{2}}} + {2{jk}\frac{\partial{E\left( {\rho,\varnothing,z} \right)}}{\delta\; z}}} = 0$Can be determined in cylindrical coordinates to be:

${E\left( {\rho,\varnothing,z} \right)} = {\sum\limits_{\ell\rho}\;{C_{\ell\rho}E_{0}\frac{w_{0}}{w(z)}\left( \frac{\sqrt{2}\rho}{w(z)} \right)^{\ell}{L_{\ell}^{\rho}\left( \frac{\sqrt{2}\rho}{w(z)} \right)}e^{{- \frac{\rho^{2}}{{w^{2}{(t)}}_{e}}} - {{j{({{2\rho} + \ell + 1})}}\tan^{{- 1}\frac{z}{z_{0}}}}}e^{j\;{\ell\varnothing}}e^{j\frac{k\;\rho^{2}}{2{R{(z)}}}}}}$The equation

$L_{\ell}^{\rho}\left( \frac{\sqrt{2}\rho}{w(z)} \right)$may also be shown as

${L_{\ell}^{\rho}\left\lbrack \frac{2\rho^{2}}{w^{2}(t)} \right\rbrack}.$

The lowest mode is the most important mode and in fact this transversemode is identical for both rectangular and cylindrical coordinates.

${\varphi\left( {\ell,{P;z}} \right)} = {\left( {{2P} + \ell + 1} \right)\tan^{- 1}\frac{z}{z_{0}}}$TEM₀₀^(rect) = TEM₀₀^(Cyl) C₀₀ = 1   H₀ = 1   L₀⁰ = 1 then$\left. {TEM}_{00}\Longrightarrow{E\left( {\rho,z} \right)} \right.\text{∼}E_{0}\frac{w_{0}}{w(z)}e^{- \frac{\rho^{2}}{w^{2}{(t)}}}e^{{- {jt{an}}^{- 1}}\frac{Z}{Z_{0}}}e^{{jk}\frac{\rho^{2}}{2{R{(z)}}}}$

Referring now more particularly to FIG. 89 , there is illustrated apatch antenna element 8410. Multiple ones of these patch antennaelements 8410 our located upon the multilayer patch antenna array 8402as discussed hereinabove. The antenna element 8410 includes a patch 8902having a length L and a width W. The patch 8902 is fed from an inputtransmission line 8904 that is connected with the feed network 8404(FIG. 84 ) and is resting upon a substrate 8906 having a height h. Themicrostrip patch antenna includes a first radiating slot 8908 along afirst edge of the patch 8902 and a second radiating slot 8910 along asecond edge of the patch 8902. The electronic field at the aperture ofeach slot can be decomposed into X and Y components as illustrated inFIG. 9 . The Y components are out of phase and cancel out because of thehalf wavelength transmission line 8904. The radiating fields can bedetermined by treating the antenna as an aperture 9000 as shown in FIG.90 having a width W 9002 and a height h 9004.

The transmission line model can be further analyzed in the followingmanner. G_(r) is the slot conductance and B_(r) is the slot susceptance.They may be determined according to the equations:

$G_{r} = \left\{ {{\begin{matrix}{{\frac{W^{2}}{90\lambda_{0}^{2}}\mspace{14mu}{for}\mspace{14mu} W} < \lambda_{0}} \\{{\frac{W}{120\lambda_{0}}\mspace{14mu}{for}\mspace{14mu} W} > \lambda_{0}}\end{matrix}B_{r}} = \frac{2{\pi\Delta\ell}\sqrt{ɛ_{eff}}}{\lambda_{0}Z_{0}}} \right.$

The input admittance of the patch antenna 8410 can be approximated as:

$Y_{in} = {Y_{slot} + {Y_{0}\frac{Y_{slot} + {{jY}_{0}{\tan\left( {\beta\left( {L + {2{\Delta\ell}}} \right)} \right)}}}{Y_{0} + {{jY}_{slot}{\tan\left( {\beta\left( {L + {2{\Delta\ell}}} \right)} \right)}}}}}$where Δl is the end effect of the microstrip.

The rectangular patch antenna 8410 will resonate when the imaginary partof the input admittance goes to zero.

The end effect may be calculated according to the equation:

${\Delta\ell} = {0.412{h\left( \frac{ɛ_{eff} + 0.3}{ɛ_{eff} - 0.258} \right)}\frac{\left( {W/h} \right) + 0.264}{\left( {W/h} \right) + 0.8}}$${L + {2{\Delta\ell}}} = {\frac{\lambda_{g}}{2} = \frac{\lambda_{0}}{2\sqrt{ɛ_{eff}}}}$$ɛ_{eff} = {\frac{ɛ_{r} + 1}{2} + {\frac{ɛ_{r} - 1}{2}\left( {1 + \frac{10h}{W}} \right)} - 0.5}$

The resonant frequency of the patch antenna 8410 is given by:

$f_{r} = \frac{C}{2\sqrt{ɛ_{eff}}\left( {L + {2{\Delta\ell}}} \right)}$Typically the width W of the aperture is given by:

$W = {\frac{C}{2f_{r}}\left( \frac{ɛ_{r} + 1}{2} \right)^{{- 1}/2}}$

The multilayered patch antenna array 8402 may transmit both HermiteGaussian beams using the processing discussed with respect to U.S.patent application Ser. No. 14/323,082 or Laguerre Gaussian beams. Whentransmitting Laguerre Gaussian beams information may be transmitted in anumber of fashions. A spiral phase plate and beam splitter approach maybe used, a dual OAM mode antenna approach may be used, or the patchedantenna described herein may be utilized. These implementations would bebeneficial in both fronthaul and backhaul applications.

In order to transmit several OAM modes of order l and amplitude a_(l)^(OAM), the antenna elements must be fed by an input signal according tothe equation:

${{a_{n}^{feed}\frac{1}{\sqrt{N}}{\sum\limits_{l = 0}^{N - 1}\;{a_{l}^{OAM}e^{{{- {j2\pi}}\frac{ln}{N}},}\mspace{14mu} n}}} \in \left\{ {0,\ldots\mspace{14mu},{N - 1}} \right\}},$

Note that the number of elements in the multilayer patch antenna array8402 limits the number of possible OAM modes due to sampling. Due toaliasing, modes of order greater than N/2 are actually modes of negativeorders.

$\left. {{{b_{I^{\prime}}^{OAM} = {{\frac{1}{\sqrt{N}}{\sum\limits_{p = 0}^{N - 1}\;{b_{p}^{feed}e^{{{j2\pi}\frac{{pI}^{\prime}}{N}},}\mspace{14mu} p}}} \in \left\{ {0,\ldots\mspace{14mu},{N - 1}} \right\}}},{h_{pn} = {{\beta e}^{- {jkr}_{np}}\frac{\lambda}{4\pi\; r_{np}}}},{r_{pn} = \sqrt{{D^{2} + R_{t}^{2} + R_{r}^{2} - {2R_{t}R_{r}{\cos\left( \theta_{np} \right)}}},}}}{\theta_{pn} = {2\pi\text{(}\frac{n - P}{N}}}} \right),\text{}{\beta = \sqrt{g_{t}g_{r}}}$Single Mode Link Budget

H_(tot) = U^(H)HUb^(OAM) = H_(tot)a^(OAM)${\frac{P_{r}}{P_{t}}(l)} = {{\frac{b_{l}^{OAM}}{a_{l}^{OAM}}}^{2} = {{\sum\limits_{p = 0}^{N - 1}\;{\sum\limits_{n = 0}^{N - 1}\;{\frac{\beta}{N}e^{- {jl\theta}_{np}}e^{- {jkr}_{np}}\frac{\lambda}{4\pi\; r_{np}}}}}}^{2}}$Asymptotic Formulation

The object is to determine an asymptotic formulation of the Link budgetat large distances, i.e. when D→+(∞), we seek the leading term for eachvalue of l Link budget −l are the same.

The link budget is asymptotically given by:

${\frac{P_{r}}{P_{t}}\left( |l| \right)} = {{\frac{\lambda\beta}{\left. {4\pi} \middle| l \middle| ! \right.}\left( \frac{{kR}_{t}R_{r}}{2} \right)^{|l|}\frac{1}{D^{|l|{+ 1}}}}}^{2}$

From the Fraunhofer distance 2 (2max(R_(t), R_(r)))²/λ=200λ, the linkbudget asymptotically tends to straight lines of slope −20 (|l|+1) dBper decade, which is consistent with an attenuation in 1/D^(2|l|+2).

Asymptotic Expressions with Gains and Free Space Losses

Gains and free space losses may be determined by:

${\frac{P_{r}}{P_{t}}\left( |1| \right)} = {\frac{{Ng}_{t}}{\left| 1 \middle| ! \right.}\left( \frac{4{\pi\left( {\pi\; R_{t}^{2}} \right)}}{\lambda^{2}} \right)^{|1|}\frac{{Ng}_{r}}{\left| 1 \middle| ! \right.}\left( \frac{4{\pi\left( {\pi\; R_{t}^{2}} \right.}}{\lambda^{2}} \right)^{|1|}\left( \frac{\lambda}{4\pi\; D} \right)^{2|1|{+ 2}}}$${L_{{FS}_{eq}}(l)} = \left( \frac{4\pi\; D}{\lambda} \right)^{2|l|{+ 2}}$${G_{eq}(l)} = {\frac{Ng}{\left| l \middle| ! \right.}\left( \frac{4{\pi\left( {\pi\; R^{2}} \right)}}{\lambda^{2}} \right)^{|l|}}$

For a fixed value of |l|, each equivalent gain increases R^(2|l|) Sothat the link budget improves by a factor of R^(4|l|). On the contrary,for a fixed value of R, when |l| increases, the link budget decreasessince asymptotically the effect of D is greater than those of R_(t) andR_(r).

Referring now to FIG. 91 , there is illustrated a 3-D model of a singlerectangular patch antenna designed for 2.42 GHz and only one linearpolarization. The radiation pattern for this antenna is illustrated inFIG. 92 .

Based on our hypothesis, the vibrational modes of COVID-19 virus can beexplored due to the dipolar mode of acoustic vibrations inside the viruswhich can be resonantly excited by microwaves of the same frequency.This is due to energy transfer from microwaves to acoustic vibrations(photon to phonon). The overall efficiency of this transfer is alsorelated to the mechanical properties of the surrounding environmentwhich influences the quality factor of the oscillation of the virus.

The virus inactivation threshold needs to be able to be measured. Theinactivation thresholds of simpler to use viruses that are close to thegeometry, size, and composition of COVID1-9 can be studied. If this is aproblem, bacteria can instead be used. The response of the sample todipolar-mode-resonance and off-resonance microwave frequencies as wellas with different microwave powers can be more fully explored. Onetechnique involves using Staph aureus or Pseudomonas aeruginosasuspended in culture medium. However, the experimental proceduresinvolve first testing with the culture medium alone and then repeat theexperiments with the bacteria in the medium so signal processing can beperformed to separate any possible contributions from the culturemedium.

To identify the mechanical vibrations, microwave resonance spectralmeasurements on the viruses must be performed. To do that, viruses areprepared where they are cultured, isolated, purified, and then preservedin phosphate buffer saline liquids at PH of 7.4 at room temperature. Ineach measurement, one microliter solution is taken by a micropipette anduniformly dropped on a coplanar waveguide apparatus. The guidedmicrowaves should be incident on the virus-containing solution. Thereflection S₁₁ and transmission S₂₁ parameters are recordedsimultaneously using a high bandwidth network analyzer (one that canmeasure from few tens of MHz to tens of GHz). The microwave attenuationspectra can be evaluated by |S₁₁|²+|S₂₁|². The attenuation spectrum ofthe buffer liquids needs to be measured with the same volume on the samedevice so the attenuation spectra of the buffer solutions can becompared with and without viruses, and deduce the microwave attenuationspectra of the viruses and identify the dominant resonance and spuriousresonances of the specific virus.

Higher inactivation of viruses at the dipolar resonant frequency can beachieved. The microwave power density threshold for a sample (COVID-19)needs to be below the IEEE safety standards for microwaves. Real-timeexperiments for reverse transcription polymerase chain reaction (RT-PCR)can confirm that the main inactivation mechanism is entirely based onphysically fracturing the viruses and the RNA genome is not impacted bythe microwave radiation.

In the IEEE Microwave Safety Standard, the spatial averaged value of thepower density in air in open public space shall not exceed theequivalent power density of 100(f_(GHz))/3)^(1/5) W/m² at frequenciesbetween 3 and 96 GHz. This corresponds to 115 W/m² at 6 GHz, 122 W/m² at8 GHz, and 127 W/m² at 10 GHz for averaged values of the power densitiesin air. Assuming all the microwave power in air transmitted into asample, and by taking the dielectric constant of water 71.92 (6 GHz),67.4 (8 GHz), and 63.04 (10 GHz) for calculation, this safety standardthen corresponds to the average electric field magnitude of 101 V/m (6GHz), 106 V/m (8 GHz), 110 V/m (10 GHz) inside the water-basedspecimens. The required threshold electric field magnitudes at theresonant frequency that rupture the viruses must be within the IEEEMicrowave Safety Standard (106 V/m), indicating high energy transferefficiency, even if the quality factor may be low. The reason forassuming water dielectric constant is that the active airborne virusesare always transported inside tiny water droplets.

The structure of the coplanar waveguide by the microfluidic channel witha sensing zone to measure the microwave absorption spectrum of samplemust be analyzed. This microwave microfluidic channel can provide a widemicrowave bandwidth (i.e. 40 GHz). The power absorption ratio by thevirus at the resonant frequency and the Q are measured by measuring thefull width at half maximum of the power spectral density. Given thedensity of viruses or bacteria in the solution, an absorption crosssection of the virus at the resonant frequency can be determined. Thethreshold electric field magnitude to fracture the virus as a functionof microwave frequency can then be estimated.

To discover the resonances, the residual infectivity of the virus afterradiating microwave of different frequency ranges are measured. In thiscase, the samples need be placed below a horn antenna. Microwaveanechoic camber or material can be used to decrease the reflection ofthe microwave. However, to check the inactivation ratio, the radiatedviruses need be analyzed by other biological methods as well to measurethe residual infectivity of viruses.

The field intensity threshold for inactivating the virus ranges betweenE₁ to E₂ V/m, which corresponds to power density Pd₁ to Pd₂ W/m², formicrowaves between f₁ to f₂ GHz. Given the aperture size of the hornantenna used, the required threshold power input ranges from P₁ to P₂Watt for f₁ to f₂ GHz microwaves can be calculated. Fixed microwavepower (higher than all the threshold power input) are first applied intothe horn antenna for studying the frequency response given thetransmission coefficient of the horn antenna. If the peak inactivationhappens at dipolar mode and at resonance the measured titer count wouldbe zero, indicating 100% inactivation ratio, which means that theremaining active viral/bacterial concentration would be smaller than thesystem sensitivity of XX pfu/mL (specify later our sample). The resultwould indicate at least a few-order of magnitudes attenuation on thevirus titer when the microwave frequency is tuned to the dipolar moderesonant frequency with the electric field intensity few times higherthan the threshold.

The microwave absorption spectrum measurement need be performed bycombining the coplanar waveguide circuit with a microfluidic channel.The gap between the signal electrode and the ground electrode of thewaveguide need be recorded. To decrease microwave loss on theelectrodes, gold layer electrodes may be used. On the surface ofelectrodes, a thermal isolator layer may be used (silicon dioxide on thesensing zone by PECVD to lower the temperature rise of fluids due tomicrowave dielectric heating). A network analyzer is used as the sourceto measure the absorption spectrum of viruses from f₁ GHz to f₂ GHz. Thespectrum of the solution without viruses is measured first as areference and then solution with viruses for comparison. By removing thesolution background (signal processing or post processing), themicrowave absorption spectrum of viruses can be measured.

For microwave radiation, a network analyzer or an YIG oscillator may beused. The microwave signal is amplified by a power amplifier and radiateit from the horn antenna. To prevent damage on oscillator and amplifierdue to back reflection, an isolator and a directional coupler are addedand everything is kept under a flow hood. The antenna should be directednormally incident on acrylic cuvettes at a short distance (i.e. 5 cmbelow the exit of horn antenna). To prevent reflection from the metalhood surface, the cuvette can be put in a plastic dish supported by abroad band pyramidal absorber like the ones used in anechoic chambers.For each measurement, the sample under radiation should be inside thecuvettes with 15 minutes at different microwave frequencies or atdifferent microwave powers. After radiation, a buffer is used to washand collect the viruses or bacteria and then the radiated solutions areused for biological study of inactivated viruses or bacteria.

A structured vector beams using patch antennas to provide the resonantfrequencies. These patch antennas radiate at frequencies as determinedabove but may be discovered by another source if needed.

Antenna Equations

The antenna signal propagation for inducing the resonance within a virus(such as COVID-19), bacterial or other organism can be modeled in thefollowing manner. The virus can be modeled as:m _(*) {umlaut over (x)}(t)+m _(*) γ{dot over (x)}(t)+m _(*)ω₀ ²x(t)=qE(t)  Virus Model:where m_(*)=effective mass

Assuming E(t)=E₀e^(−iωt) define harmonic oscillations as:

x(t) = X₀e^(−i ω t)∼X₀cos ω t${\overset{.}{x}(t)} = {{{- i}\;\omega\; X_{0}e^{{- i}\;\omega\; t}} = {{- i}\;\omega\; x}}$${\overset{¨}{x}(t)} = {{\left( {{- i}\;\omega} \right)^{2}X_{0}e^{{- i}\;\omega\; t}} = {{{- \omega^{2}}\; X_{0}e^{{- i}\;\omega\; t}} = {{{{- \omega^{2}}x} - {m_{*}\omega^{2}X_{0}e^{{- i}\;\omega\; t}} - {i\; m_{*}{\gamma\omega}\; X_{0}e^{i\;\omega\; t}} + {m_{*}\omega_{0}^{2}X_{0}e^{{- i}\;\omega\; t}}} = {{qE}_{0}e^{{- i}\;\omega\; t}}}}}$$X_{0} = {{\frac{\frac{{qE}_{0}}{m_{*}}}{\omega_{0}^{2} - \omega^{2} - {i\;{\gamma\omega}}} - {m_{*}\omega^{2}ϰ} - {i\; m_{*}{\gamma\omega ϰ}} + {m_{*}\omega^{2}ϰ}} = {{0 - {m_{*}\omega^{2}} + {i\;\omega\;\gamma\; m_{*}} + {m\;\omega_{0}^{2}}} = 0}}$$\omega = \frac{{{- i}\;\gamma\; m_{*}} \pm \sqrt{{{- \gamma^{2}}m_{*}^{2}} + {4{km}_{*}}}}{{- 2}m_{*}}$$\omega = \frac{{i\;\gamma\; m_{*}} \pm \sqrt{{{- \gamma^{2}}m_{*}^{2}} + {4{km}_{*}}}}{2m_{*}}$

The Decay rate=imaginary part of ω

$\frac{{\gamma m}_{*}}{2m_{*}} = \frac{\omega_{0}}{2Q}$Damping

$\begin{matrix}{\gamma = \frac{\omega_{0}}{Q}} \\{k = {{m_{*}\omega_{0}^{2}\omega_{0}} = \sqrt{\frac{k}{m_{*}}}}} \\{{❘X_{0}❘} = {\frac{\frac{{qE}_{0}}{m_{*}}}{\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2} + \left( {- {\gamma\omega}} \right)^{2}}} = \frac{\frac{{qE}_{0}}{m_{*}}}{\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right) + {\gamma^{2}\omega^{2}}}}}} \\{{❘X_{0}❘} = \frac{\frac{{qE}_{0}}{m_{*}}}{\sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2} + \left( \frac{{\omega\omega}_{0}}{Q} \right)^{2}}}} \\{\gamma = {{\frac{\omega_{0}}{Q}Q} = \frac{\omega_{0}}{\gamma}}} \\{\left. {\angle X}_{0}\rightarrow{\tan\phi} \right. = {{\frac{{\omega\omega}_{0}}{Q\left( {\omega_{0}^{2} - \omega^{2}} \right)}\phi} = {\tan^{- 1}\frac{{\omega\omega}_{0}}{Q\left( {\omega_{0}^{2} - \omega^{2}} \right)}}}} \\{{x(t)} = {{❘X_{0}❘}{\cos\left( {{\omega t} + \phi} \right)}}}\end{matrix}$

The absorption model for determining the absorption of radiating signalsfrom an antenna may be determined in the following manner. TheAbsorption Model is defined by:

$\begin{matrix}{{{\epsilon_{r}\epsilon_{0}} + \frac{i\sigma}{\omega}} = \epsilon} \\{\delta = {\frac{1}{\omega}\left\lbrack {\frac{\mu\epsilon}{2}\left( {\sqrt{1 + \left( \frac{\sigma}{\omega\epsilon} \right)^{2}} - 1} \right)} \right\rbrack}^{\frac{1}{2}}} \\\begin{matrix}{f = {{1{GHz}\delta} = {3.4{cm}}}} \\{f = {{10{GHz}\delta} = {0.27{cm}}}}\end{matrix}\end{matrix}$

The Specific absorption rate (SAR) is defined by:

${SAR} = {{\frac{d}{dt}\left\lbrack \frac{dW}{dm} \right\rbrack} = {\frac{d}{dt}\left\lbrack \frac{dW}{pdv} \right\rbrack}}$

Energy absorption is defined by:

W=energy

${SAR} = {\frac{\left. \sigma \middle| E \right|^{2}}{\rho}\begin{matrix} \\

\end{matrix}}$

The Poynting vector {right arrow over (P)} is power density of EM Powerflow/unit area. The radiated power crossing a surface d{right arrow over(s)} is for an Isotropic case:

$\begin{matrix}{{dW}_{tx} = {\left. {{\overset{\rightarrow}{P} \cdot d}\overset{\rightarrow}{s}}\rightarrow w \right. = {{∯{{\overset{\rightarrow}{P} \cdot d}\overset{\rightarrow}{s}}} = {{∯{P_{r}{ds}}} = {P_{r}{\overset{\pi 2\pi}{∯\limits_{\theta = {{0\phi} = 0}}}{r^{2}\sin\theta{d\theta d\phi}}}}}}}} \\{{{Radial}{Part}{of}{the}{Poynting}{Vector}{or}{Power}{Density}P_{r}} = {\frac{W_{tx}}{4\pi r^{2}}\frac{W}{m^{2}}}} \\{\left. {{Radiation}{intensity}U}\rightarrow{{power}{per}{unit}{s{olid}}{angle}\frac{W}{4\pi}} \right.}\end{matrix}$For isotropic U=U₀

$U_{0} = {{r^{2}P_{r}} = \frac{W}{4\pi}}$

The directivity of various types of antennas can then be defined by:

$\begin{matrix}{{{Directivity}{of}{antenna}D} = {\frac{{maximum}{radiated}{intensity}}{{intensity}{radiated}{by}{isotropic}} = {{{}\frac{U_{\max}}{U_{0}}}}}} \\{{{Directivity}{of}{isotropic}{antenna}D_{0}} = {{\frac{{maximum}{radiated}{intensity}}{{intensity}{radiated}{by}{isotropic}} = {{{{{}\frac{U_{\max}}{U_{0}}}}}}}}} \\{{{Directivity}{of}a{hemispherical}{antenna}D_{h}} = {\frac{2U_{o}}{U_{o}} = 2}}\end{matrix}$

Therefore

$\begin{matrix}{{D{is}{also}D} = \frac{{maximum}{radiated}{intensity}}{{average}{radiated}{intensity}}} \\{D = {\frac{U_{\max}}{\frac{W}{4\pi}} = {\frac{4\pi U_{\max}}{W} = {{\frac{4\pi}{B}B} = {\frac{W}{U_{\max}}{beam}{area}}}}}} \\{D = {\frac{U_{\max}}{r^{2}P_{r}} = {{\frac{4\pi}{B}B} = {\frac{W}{U_{\max}} = {\frac{∯{P_{r}{ds}}}{U_{\max}} = {\frac{∯{\frac{U}{r^{2}}{ds}}}{U_{\max}} = \frac{∯{Ud\Omega}}{U_{\max}}}}}}}}\end{matrix}$

The Total power W within a conical region is:

W = ∫₀^(2π)∫₀^(θ)U_(max)sin θ^(′)dθ^(′)dϕ = 2π(1 − cos θ)U_(max)

Therefore

${D = {\frac{4\pi}{2{\pi\left( {1 - {\cos\theta}} \right)}} = \frac{W}{U_{\max}}}}\begin{matrix}{{{For}{hemisphere}\theta} = {{\frac{\pi}{2}D_{hemi}} = 2}} \\{{{For}a{complete}{sphere}\theta} = {{\pi D_{sphere}} = {1{isotropic}}}} \\{{{{Approximate}{beam}{area}} = {B = {{\Delta\theta\Delta\phi}{\Delta\theta}}}},{{\Delta\phi}{are}{half}–{beam}{widths}}}\end{matrix}$

The Gain of an antenna can be determined according to:G=ηD

Therefore power density is determined according to:

$\begin{matrix}{P_{r} = \frac{G_{t}W_{t}}{4\pi r^{2}}} \\{W_{t} = \frac{\left( {4\pi r^{2}} \right)P_{r}}{G_{t}}}\end{matrix}$

For a uni-directional cosine power source:

$\begin{matrix}{U = {U_{\max}\cos\theta}} \\{W = {{\int_{0}^{2\pi}{\int_{0}^{\frac{\pi}{2}}{U_{\max}\cos{\theta sin}\theta d\theta d\phi}}} = {\pi U_{\max}}}} \\{D = {\frac{4\pi}{2{\pi\left( {1 - {\cos\theta}} \right)}} = {\frac{4\pi}{2{\pi\left( {1 - {\cos 60{^\circ}}} \right)}} = {\frac{4\pi}{\pi} = {{4\theta} = {60{^\circ}}}}}}} \\{D = {\frac{4\pi}{2{\pi\left( {1 - {\cos\theta}} \right)}} = {\frac{4\pi}{2{\pi\left( {1 - {\cos 90{^\circ}}} \right)}} = {\frac{2\pi}{\pi} = {{2\theta} = {90{^\circ}}}}}}}\end{matrix}$

According to IEEE safety standards:

$P_{r}^{IEEE} = {100\left( \frac{f_{GHz}}{3} \right)^{\frac{1}{5}}{W/m^{2}}}$

With 50% antenna efficiency η=½

$\begin{matrix}{G = {{\eta D} \approx \frac{D}{2}}} \\{P_{r} = {\frac{\left( \frac{D}{2} \right)W_{t}}{4\pi r^{2}}\frac{W}{m^{2}}}}\end{matrix}$

To be lower than IEEE safety standard P_(r) ^(IEEE) by 10%:

$\begin{matrix}{P_{r_{c}} = {\left( {90\%} \right)100\left( \frac{f}{3} \right)^{\frac{1}{5}}}} \\{\frac{\left( \frac{D}{2} \right)W_{t}}{4\pi r^{2}} = {90\left( \frac{f_{GHz}}{3} \right)^{\frac{1}{5}}}} \\{W_{t} = {90\left( \frac{f_{GHz}}{3} \right)^{\frac{1}{5}}\frac{4\pi r^{2}}{\frac{D}{2}}{Watts}}} \\{{{For}D} = {{2\theta} = {90{^\circ}}}} \\{W_{t}^{\frac{\pi}{2}} = {90\left( \frac{f_{GHz}}{3} \right)^{\frac{1}{5}}4\pi r^{2}}} \\{{{For}D} = {{4\theta} = {60{^\circ}}}} \\{W_{t}^{\frac{\pi}{3}} = {90\left( \frac{f_{GHz}}{3} \right)^{\frac{1}{5}}2\pi r^{2}}}\end{matrix}$

In general:

${W_{t} = {\frac{4\pi r^{2}P_{r}}{G_{t}} = {\frac{4\pi r^{2}P_{r}}{\eta D} = {{{For}U} = {U_{\max}\cos\theta{cosine}{power}{source}}}}}}\text{}{D = \frac{2}{1 - {\cos\theta}}}{W_{t}^{\theta} = {\frac{4\pi r^{2}P_{r}}{\eta\left( \frac{2}{1 - {\cos\theta}} \right)} = \frac{4\pi{r^{2}\left( {1 - {\cos\theta}} \right)}P_{r}}{2\eta}}}{W_{t}^{\theta} = {\frac{2\pi r^{2}}{\eta}\left( {1 - {\cos\theta}} \right)P_{r}}}{W_{t}^{\theta} = {\frac{2\pi r^{2}}{\eta}{\left( {1 - {\cos\theta}} \right)\left\lbrack {90\left( \frac{f_{GHz}}{3} \right)^{\frac{1}{5}}} \right\rbrack}}}$Microwave Resonance

There has been a lot of work done on the influenza virus and themeasurements of parameters done on the influenza virus can be used toestimate some of the parameters for Covid-19. In general, microwaveresonance absorption measurements are performed to identify themechanical resonance of the virus/bacteria as shown in FIG. 93 .Initially, the reflection S₁₁ and transmission S₁₂ parameters must bemeasured simultaneously with a large bandwidth network analyzer at step9302. The attenuation spectra is plotted at step 9304 as a function offrequency (GHz) attenuation due to absorption, reflection andtransmission. This is more particularly illustrated in FIG. 94 . Asshown in FIG. 95 , the background attenuation spectra is shown generallyat 9402. The overall background attenuation is shown generally at 9404.

Various absorption related terms are defined in the following manner:

${{{Normalized}{insertion}{loss}} = {1 - \frac{A(f)}{A_{b}(f)}}}{{A(f)} = {{Absorption}{spectra}}}{{A_{b}(f)} = {{non} - {resonant}{background}{absorption}}}$This can be plotted as displayed in FIG. 95 .

The resonance frequency can be defined as:

${\omega_{0} = {2\pi f_{0}{resonance}{frequency}}}{Q = {\frac{f_{0}}{\Delta f} = \frac{\omega_{0}}{\Delta\omega}}}$

The plot with respect to the resonance frequency is shown in FIG. 95 .The dominate dipole mode 9502 occurs at frequency f_(o) and the highermode 9504 occurs at frequency 2f_(o)

As described herein above, the equation for the Electric field necessaryto rupture the capsid of the virus/bacteria is defined by:

$E = {\frac{c_{2}}{c_{1}}P_{stress}^{avg}{\frac{\pi r^{2}}{q\omega_{0}^{2}}\left\lbrack \sqrt{\left( {\omega_{0}^{2} - \omega^{2}} \right)^{2} + \left( \frac{{\omega\omega}_{0}}{Q} \right)^{2}} \right\rbrack}}$where ω=2πf is the microwave radiation frequencyω₀=the resonance frequency of the virus/bacteriar=radius of the virus/bacteriaq=charge of the virus/bacteriaP_(stress) ^(avg)=average stress to rupture the virus/bacteriac₁=constant 1 P_(stress) ^(max)=c₁*P_(stress) ^(avg)c₂=constant 2 c₂=% shell region of equatorial plane

From microwave resonance absorption measurements, ω₀ and Q areavailable.

-   -   r=known

For influenza H3N2, the following values have been determined:

${c_{1} = 2},{c_{2} = 0.58},{P_{stress}^{avg} = {0.141 \times 10^{6}\frac{N}{m^{2}}}},{q = {1.16 \times 10^{7}e}}$Shell area=A_(s)Total equatorial area=A_(T)Core area=A_(c)

${A_{t} = {\pi r_{0}^{2}}}{A_{c} = {\pi r_{i}^{2}}}{A_{s} = {\pi\left( {r_{0}^{2} - r_{i}^{2}} \right)}}{\frac{A_{s}}{A_{t}} = {\frac{\pi\left( {r_{0}^{2} - r_{i}^{2}} \right)}{\pi r_{0}^{2}} = {\frac{r_{0}^{2} - r_{i}^{2}}{r_{0}^{2}} = {{1 - \frac{r_{i}^{2}}{r_{0}^{2}}} = {{1 - \left( \frac{r_{i}}{r_{0}} \right)^{2}} = c_{2}}}}}}{{1 - c_{2}} = \left( \frac{r_{i}}{r_{0}} \right)^{2}}{\frac{r_{i}}{r_{0}} = \sqrt{1 - c_{2}}}{r_{i} = {\sqrt{1 - c_{2}}r_{0}}}{{E│_{\omega = \omega_{0}}} = {\frac{\beta}{\alpha}P\frac{\pi r^{2}}{qQ}}}$

The Charge q can be estimated from absorption measurements. Fromabsorption cross section σ_(abs):

$\sigma_{abs} = {{{- \frac{1}{N*L}}{\ln\left( {1 - \alpha} \right)}} = {2.5 \times 10^{- 13}m^{2}}}$

In influenza measurementsN=7.5×10¹⁴ l/m³L=1.25×10⁻³ mα=21%=0.21

However, theoretical absorption cross-section of virus/bacteria atresonant frequency is:

${\sigma_{abs} = \frac{{Qq}^{2}}{\omega_{0}m_{*}c\sqrt{\epsilon_{r}}\epsilon_{0}}}{\epsilon_{r} = {{permitivity}{of}{phosphate}{buffer}{saline}}}{q^{2} = {\frac{\sigma_{abs}\omega_{0}m_{*}c\sqrt{\epsilon_{r}}\epsilon_{0}}{Q} = {1.16 \times 10^{7}e}}}{{for}{influenza}H3N2}$

The reduced or effective mass m, must then be found. For influenza H3N2most of the mass is concentrated in shell of the virus.m _(virus)=161 MDa=161×10⁶ Dam _(shell)=90% of m _(virus)m _(core)=10% of m _(virus)

The virus shell has lipid, neuraminidase (NA), hemagglutinin (HA) andM-Proteins. The virus core has RNA, RNP (ribonucleoproteins). Because ofthe spring-mass model of the core and shell the reduced mass m_(*) isfound from:

${\frac{1}{m_{*}} = {{\frac{1}{m_{s}} + \frac{1}{m_{c}}} = {\frac{m_{s} + m_{c}}{m_{s}m_{c}} = \frac{m_{virus}}{m_{s}m_{c}}}}}{m_{*} = {\frac{m_{s}m_{c}}{m_{virus}} = \frac{(0.9){m_{v}(0.1)}m_{v}}{m_{v}}}}{m_{*} = {{0.09m_{v}} = {14.5 \times 10^{6}{Da}}}}{{1{Da}} = {1.66054 \times 10^{- 21}{kg}}}$

The information needed to calculate the magnitude of electric field Eexcept P_(stress) ^(avg) that is needed to rupture the capsid of thevirus/bacteria and ensure it is below the IEEE safety standard is nowavailable based upon this data. The German paper by Sai Li & FredericEghiaian, “Bending & Puncturing the Influenza lipid Envelope” BiophysJ., Feb. 2, 2011; 100(3): 637-645, which is incorporated herein byreference in its entirety, covers the exact Atomic Force Microscopy(AFM) measurements to show that:P _(stress) ^(avg)=0.141×10⁶ N/m²The minimum of the Electric field can be determined according to:

${{{Let}x} = \frac{\omega}{\omega_{0}}}{\frac{dE}{dx} = \frac{{2Q^{2}x^{3}} - {2Q^{2}x} - x}{Q^{2}\sqrt{{\left( {\frac{1}{Q^{2}} - 2} \right)x^{2}} + x^{4} + 1}}}{\frac{dE}{d\omega} = {{\frac{1}{\omega_{0}}\frac{dE}{dx}} = {\left\lbrack {\frac{1}{\omega_{0}}\frac{{2{Q^{2}\left( \frac{\omega}{\omega_{0}} \right)}^{3}} - {2{Q^{2}\left( \frac{\omega}{\omega_{0}} \right)}} + \frac{\omega}{\omega_{0}}}{Q^{2}\sqrt{{\left( {\frac{1}{Q^{2}} - 2} \right)\left( \frac{\omega}{\omega_{0}} \right)^{2}} + \left( \frac{\omega}{\omega_{0}} \right)^{4} + 1}}} \right\rbrack C_{1}}}}{\frac{dE}{d\omega} = {\frac{\beta}{\alpha}P_{stress}^{avg}{\frac{\pi r^{2}}{q}\left\lbrack {\frac{1}{\omega_{0}}\frac{{2{Q^{2}\left( \frac{\omega}{\omega_{0}} \right)}^{3}} + {\left( \frac{\omega}{\omega_{0}} \right)\left\lbrack {1 - {2Q^{2}}} \right\rbrack}}{Q^{2}\sqrt{{\left( {\frac{1}{Q^{2}} - 2} \right)\left( \frac{\omega}{\omega_{0}} \right)^{2}} + \left( \frac{\omega}{\omega_{0}} \right)^{4} + 1}}} \right\rbrack}}}\text{}{\frac{dE}{d\omega} = 0}{{{2{Q^{2}\left( \frac{\omega}{\omega_{0}} \right)}^{3}} + {\frac{\omega}{\omega_{0}}\left\lbrack {1 - {2Q^{2}}} \right\rbrack}} = 0}{{{2{Q^{2}\left( \frac{\omega}{\omega_{0}} \right)}^{2}} + \left\lbrack {1 - {2Q^{2}}} \right\rbrack} = 0}{{2{Q^{2}\left( \frac{\omega}{\omega_{0}} \right)}^{2}} = {{2Q^{2}} - 1}}{\left( \frac{\omega}{\omega_{0}} \right) = \sqrt{\frac{{2Q^{2}} - 1}{2Q^{2}}}}{\omega = {\sqrt{\frac{{2Q^{2}} - 1}{2Q^{2}}}\omega_{0}}}$

FIGS. 96-100 illustrates power density matlab plots for the aboveantenna equations. FIG. 96 illustrates the maximum transmit powerrequired to achieve the IEEE safety power density threshold for 1 GHz, 5GHz, 10 GHz and 15 GHz for theta=30. FIG. 97 illustrates the maximumtransmit power required to achieve the IEEE safety power densitythreshold for 1 GHz, 5 GHz, 10 GHz and 15 GHz for theta=60. FIG. 98illustrates the maximum transmit power required to achieve the IEEEsafety power density threshold for 1 GHz, 5 GHz, 10 GHz and 15 GHz fortheta=90.

As described above the equation for the incident electric field (E) atthe resonance frequency of the virus or bacteria. To calculate theminimum field necessary to rupture a capsid, the derivative of theminimum of the electric field verses angular frequency over distance(dE/dw) is taken and set to zero to calculate the minimum filed neededat the radiator antenna. FIGS. 99 a and 99 b illustrate a plot of dE/dwversus distance. FIG. 100 illustrates a plot of the minimum of theelectric field (Emin) needed at the radiator antenna as a function ofdistance. The Emin needed at the antenna is larger as you go away fromthe antenna to achieve rupturing of the capsid.

FIG. 101 illustrates various manners for implementing an antenna fortransmitting the signals as described herein through an antenna 9602plugging into an electrical socket, an antenna 9604 mounted on aceiling, an antenna 9606 screwing into a light socket or an antenna 9608within a flashlight. The antennas which each include the structuresdescribed herein and provide a manner for radiating a virus, bacteria orother biological entity with a signal for inducing resonance therein asdescribed above.

The determination of the frequency to rupture the capsid virus may bedetermined in one manner according to the following. The frequency ofvibration of the virus or other biological material is a function ofYoungs modulus of elasticity, density and Poisson ratio as well as theradius and as a unitless parameter from shell model of vibration inspherical coordinates. Starting from 3D Hooke's law, the stress andstrain on the virus or biological material may be determined accordingto the normal stress and tangential stress as shown below:

Normal Stressσ_(x)=2μϵ_(xx)+λ(ϵ_(xx)+ϵ_(yy)+ϵ_(zz))Tangential stressτ_(xy)=μϵ_(xy)Where μ and λ are Lame's coefficients

${\mu = {\frac{E}{2\left( {1 + v} \right)} = {G{shear}{modules}}}}{\lambda = \frac{Ev}{\left( {1 + v} \right)\left( {1 - {2v}} \right)}}$Where E is modulus of Elasticity and ν is Poisson coefficient then themotion of elastic body is described by Navier's equation.

${{\mu{\nabla^{2}\overset{\rightarrow}{u}}} + {\left( {\lambda + \mu} \right){\nabla\left( {\nabla{\cdot \overset{\rightarrow}{u}}} \right)}} + {\rho\overset{\rightarrow}{f}}} = {\rho\frac{\delta^{2}\overset{\rightarrow}{u}}{\delta t^{2}}}$u=u({right arrow over (x)}, t) displacementf body forcesρ densityλ, μ Lame's constants{right arrow over (f)}=∇f+∇×F

-   -   f=scalar potential F=vector potential

Also{right arrow over (u)}=∇Φ+∇×{right arrow over (Ψ)} Φ=scalar potential{right arrow over (Ψ)}=vector potential

Then

${{\nabla\left( {{c_{1}^{2}{\nabla^{2}\Phi}} + f - \frac{\partial^{2}\Phi}{\partial t^{2}}} \right)} + {\nabla{\times \left( {{c_{2}^{2}{\nabla^{2}\Psi}} + \overset{\rightarrow}{F} - \frac{\partial^{2}\Psi}{\partial t^{2}}} \right)}}} = 0$Where longitudinal wave velocity

${c_{1} = \sqrt{\frac{\lambda + {2\mu}}{\rho}}}{c_{2} = \sqrt{\frac{\mu}{\rho}}}$shear wave velocity

The Navier's equation is satisfied if:

${{c_{1}^{2}{\nabla^{2}\Phi}} + f - \frac{\partial^{2}\Phi}{\partial t^{2}}} = 0$And

${{c_{2}^{2}{\nabla^{2}\Psi}} + F - \frac{\partial^{2}\Psi}{\partial t^{2}}} = 0$

In spherical coordinates:Ψ=rΨ{circumflex over (r)}+

∇×(rχ{right arrow over (r)})Ψ, χ scalar

=length factor

Therefore, with body forces neglected:c ₁ ²∇²Φ={umlaut over (Φ)}c ₂ ²Π²Ψ={umlaut over (Ψ)}c ₂ ²∇²χ={umlaut over (χ)}

Then displacement components are:

${u_{r} = {\frac{\partial\Phi}{\partial r} + {\ell\left\lbrack {\frac{\partial^{2}\left( {r\chi} \right)}{\partial r^{2}} - {r{\nabla^{2}\chi}}} \right\rbrack}}}{u_{\theta} = {{\frac{1}{r}\frac{\partial\Phi}{\partial\theta}} + {\frac{1}{sin\theta}\frac{\partial\Psi}{\partial\phi}} + {\frac{1}{r}\frac{\partial^{2}\left( {r\chi} \right)}{{\partial\theta}{\partial r}}}}}{u_{\phi} = {{\frac{1}{r{sin\theta}}\frac{\partial\Phi}{\partial\phi}} - {\frac{\partial\Psi}{\partial\phi}\frac{1}{r{sin\theta}}\frac{\partial^{2}\left( {r\chi} \right)}{{\partial\phi}{\partial r}}}}}$

Therefore stress

$\sigma_{rr} = {{\lambda{\nabla^{2}\phi}} + {2\mu\frac{\partial^{2}\Phi}{\partial r^{2}}} + {2{\mu\ell}{\frac{\partial}{\partial r}\left\lbrack {\frac{\partial^{2}\left( {r\chi} \right)}{\partial r^{2}} - {r{\nabla^{2}\chi}}} \right\rbrack}}}$

Similarly, we can find:σ_(θθ),σ_(ϕϕ),σ_(rθ),σ_(rϕ),σ_(θϕ)

Using separation of variables, the Helmholtz equation can be solved inspherical coordinates (r, θ, ϕ) with:

Bessel equation for r

Legendre equation for θ

Simple 2^(nd) order equation for ϕ

Helmholtz equation

${{\nabla^{2}f} - {\frac{1}{c^{2}}\frac{\partial^{2}f}{\partial t^{2}}}} = 0$

Expanding the Laplacian in spherical coordinates provides:

${{\frac{1}{r^{2}}{\frac{\partial}{\partial r}\left( {r^{2}\frac{\partial f}{\partial r}} \right)}} + {\frac{1}{r^{2}\sin\theta}{\frac{\partial}{\partial\theta}\left( {\sin\theta\frac{\partial f}{\partial\theta}} \right)}} + {\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}f}{\partial\phi^{2}}} - {\frac{1}{c^{2}}\frac{\partial^{2}f}{\partial t^{2}}}} = 0$Wheref(r,θ,ϕ,t)=F ₁(r)F ₂(θ)F ₃(ϕ)e ^(−iωt)

Therefore equation

$\begin{matrix}{{{r^{2}\frac{d^{2}F_{1}}{{dr}^{2}}} + {2r\frac{{dF}_{1}}{dr}} + {\left( {{k^{2}r^{2}} - P^{2}} \right)F_{1}}} = 0} & {\theta{equation}}\end{matrix}$ $\begin{matrix}{{{\frac{1}{sin\theta}\frac{d}{d\theta}\left( {{sin\theta}\frac{{dF}_{2}}{d\theta}} \right)} + {\left( {P^{2} - \frac{q^{2}}{\sin^{2}\theta}} \right)F_{2}}} = 0} & \end{matrix}$ $\begin{matrix}{{\frac{d^{2}F_{3}}{{d\phi}^{2}} + {q^{2}F_{3}}} = 0} & {\phi{equation}}\end{matrix}$Where

$k = \frac{\omega}{c}$and P², q² are separation constants

${P^{2} = {v\left( {v + 1} \right)}}{\mu = {\cos\theta}}{F_{1} = \frac{R(r)}{\sqrt{kr}}}$

Then the

$\begin{matrix}{{{r^{2}\frac{d^{2}R}{{dr}^{2}}} + {2r\frac{dR}{dr}} + {\left( {{k^{2}r^{2}} - \left( {v + \frac{1}{2}} \right)^{2}} \right)R}} = {0{Bessel}{}{equation}}} & {r{equation}}\end{matrix}$ $\begin{matrix}{{{\left( {1 - \mu^{2}} \right)\frac{d^{2}F_{2}}{{d\mu}^{2}}} - {2\mu\frac{{dF}_{2}}{d\mu}} + {\left\lbrack {{v\left( {v + 1} \right)} - \frac{m^{2}}{1 - \mu^{2}}} \right\rbrack F_{2}}} = {0{Legendre}{equation}}} & {\theta{equation}}\end{matrix}$

Therefore, the final solution is:

${f\left( {r,\ \theta,\ \phi,\ t} \right)} = {{{e^{{- i}\omega t}\text{⁠}\frac{1}{\sqrt{kr}}}}\text{⁠}{\left\lbrack {{A{J_{n + \frac{1}{2}}\left( {kr} \right)}} + \text{⁠}{{}\text{⁠}{{{\left. {{BY}_{n + \frac{1}{2}}({kr})} \right\rbrack\left\lbrack {{c{P_{n}^{m}(\mu)}} + {D{Q_{n}^{m}(\mu)}}} \right\rbrack}\left\lbrack {{Ee^{{im}\phi}} + {Fe^{- {{im}\phi}}}} \right\rbrack}}}} \right.}}$WhereP_(n) ^(m)(μ), Q_(n) ^(m)(μ) associated Legendre functions

${J_{n + \frac{1}{2}}({kr})}\mspace{14mu}{is}\mspace{14mu}{Bessel}\mspace{14mu}{function}\mspace{14mu}{of}\mspace{14mu}{first}\mspace{14mu}{kind}$${Y_{n + \frac{1}{2}}({kr})}\mspace{14mu}{is}\mspace{14mu}{Bessel}\mspace{14mu}{function}\mspace{14mu}{of}\mspace{14mu}{second}\mspace{14mu}{kind}$Q_(n) ^(m) is singular at μ=±1 so we exclude it

Now the potentials are:

${\Phi = {{Z_{n}^{(i)}\left( {\alpha r} \right)}{P_{n}^{m}\left( {\cos\theta} \right)}e^{i({{m\phi} - {\omega t}})}}}{\Psi = {{Z_{n}^{(i)}\left( {\beta r} \right)}{P_{n}^{m}\left( {\cos\theta} \right)}e^{i({{m\phi} - {\omega t}})}}}{\chi = {{Z_{n}^{(i)}\left( {\beta r} \right)}{P_{n}^{m}\left( {\cos\theta} \right)}e^{i({{m\phi} - {\omega t}})}}}{where}{\alpha = \frac{\omega}{c_{1}}}{\beta = \frac{\omega}{c_{2}}}{Z_{n}^{(1)} = {{j_{n}({kr})} = {{\sqrt{\frac{\pi}{2{kr}}}{J_{n + \frac{1}{2}}({kr})}Z_{n}^{(2)}} = {{y_{n}({kr})} = {\sqrt{\frac{\pi}{2{kr}}}{Y_{n + \frac{1}{2}}({kr})}}}}}}$

The values of μ_(r), μ_(θ), μ_(ϕ) may then be found as functions of thespherical Bessel functions. The values of σ_(rr), σ_(θθ), σ_(ϕϕ),σ_(rθ), σ_(rϕ), σ_(θϕ) can also be found as functions of Φ, Ψ, χ. Themodal characteristic equations can then now be derived.

There are two cases for a vibrating sphere. These include 1) anAxi-symmetric case (symmetry axis θ=0) no ϕ component a, and 2)Traction-free boundary conditions.

With respect to the Axi-symmetric case:

Symmetry axis θ=0

${u_{\phi} = 0}{\frac{\partial}{\partial\phi} = 0}{q = 0}{F_{3} = {constant}}{m = 0}\left. {P_{n}^{m}(\mu)}\rightarrow{P_{n}(\mu)} \right.$Therefore

$\begin{matrix}{\Phi = {{Z_{n}^{(i)}\left( {\alpha r} \right)}{P_{n}({cos\theta})}e^{- {i\omega t}}}} & {u_{r} = {\frac{\partial\Phi}{\partial r} + {\ell\left\lbrack {\frac{\partial^{2}({rx})}{\partial r^{2}} - {r{\nabla^{2}\chi}}} \right\rbrack}}} \\{\Psi = {{Z_{n}^{(i)}\left( {\beta r} \right)}e^{- {i\omega t}}}} & {u_{\theta} = {{\frac{1}{r}\frac{\partial\Phi}{\partial\theta}} + {\frac{1}{r}\frac{\partial^{2}\left( {r\chi} \right)}{{\partial\phi}{\partial r}}}}} \\{\chi = {{Z_{n}^{(i)}\left( {\beta r} \right)}{P_{n}({cos\theta})}e^{- {i\omega t}}}} & {u_{\phi} = 0}\end{matrix}$

The values of σ_(rr), σ_(θθ), σ_(ϕϕ), σ_(rθ) are can also be found andσ_(rϕ)=0σ_(θϕ)=0

Substituting Φ, Ψ, χ into stress components referenced above enables thefinding of stresses in terms of Bessel and Legendre polynomials

Traction Boundary Conditions

Boundary conditions of inner & outer surfaces of sphere becomeσ_(rr)=σ_(rϕ)=σ_(rθ)=0 at r=r _(i)σ_(rr)=σ_(rϕ)=σ_(rθ)=0 at r=r ₀

New stress equations can be constructed satisfying the above boundaryconditions. Therefore, modal characteristic equation becomes:

$\Delta_{n} = {\left| \begin{matrix}{T_{11}^{(1)}\left( {\alpha r}_{i} \right)} & {T_{11}^{(2)}\left( {\alpha r}_{i} \right)} \\{T_{11}^{(1)}\left( {\alpha r}_{0} \right)} & {T_{11}^{(2)}\left( {\alpha r}_{0} \right)}\end{matrix} \right| = {{0{for}n} = 0}}$WhereT ₁₁ ^((i))(αr)=(n ² −n−½β² r ²)Z _(n) ^((i))(αr)+2αrZ _(n+1) ^((i))(αr)Kirchhoff-Love Shell theory

Assuming that:h<<a h=shell thicknessa=radius of midsurfaceu _(r) <<hu _(θ) <<h displacement<<thicknessu _(ϕ) <<hσrr≈0 negligibleThereforeDisplacement vector: u_(r)=U_(r): displacement of middle surface

$u_{\theta} = {{\left( {1 + \frac{x}{a}} \right)u_{\theta}} - {\frac{x}{a}\frac{\partial u_{r}}{\partial\theta}}}$Wherex=r−a U _(r) and U _(θ) are functions of θ only

Then kinetic energy

$T = {\frac{1}{2}\rho_{s}{\int_{0}^{2\pi}{\int_{0}^{\pi}{\int_{- \frac{h}{2}}^{\frac{h}{2}}{\left( {{\overset{.}{u}}_{r}^{2} + {\overset{.}{u}}_{\theta}^{2}} \right)\left( {a + x} \right)^{2}\sin\;\theta\;{dxd}\;\theta\; d\;\phi}}}}}$

Neglecting x in comparison to midsurface radius a

$T = {\frac{1}{2}\rho_{s}{\int_{0}^{2\pi}{\int_{0}^{\pi}{\int_{- \frac{h}{2}}^{\frac{h}{2}}{\left( {{\overset{.}{u}}_{r}^{2} + {\overset{.}{u}}_{\theta}^{2}} \right)a^{2}\sin\theta{dxd}\;\theta\; d\;\phi}}}}}$

Neglecting rotatory inertia provides

$T = {\pi\rho_{s}ha^{2}{\int_{0}^{2\pi}{\left( {{\overset{.}{U}}_{r}^{2} + {\overset{.}{U}}_{\theta}^{2}} \right)\sin\theta d\theta}}}$

And therefore, the strain components become

$\epsilon_{\theta\theta} = {\frac{1}{a + x}\left( {\frac{\partial u_{\theta}}{\partial\theta} + u_{r}} \right)}$$\epsilon_{\phi\phi} = {\frac{1}{a + x}\left( {{\cot\theta u_{\theta}} + u_{r}} \right)}$

Therefore

$\epsilon_{\theta\theta} = {{{\frac{1}{a + x}\left( {\frac{\partial U_{\theta}}{\partial\theta} + U_{r}} \right)} + {\frac{x}{a\left( {a + x} \right)}\left( {\frac{\partial U_{\theta}}{\partial\theta} - \frac{\partial^{2}U_{r}}{\partial\theta^{2}}} \right)\epsilon_{\phi\phi}}} = {{\frac{1}{a + x}\left( {{\cot\theta U_{\theta}} + U_{r}} \right)} + {\frac{x}{a\left( {a + x} \right)}\cot{\theta\left( {u_{\theta} - \frac{\partial U_{r}}{\partial\theta}} \right)}}}}$

Following the same approach for strain potential energy the finalequations of motions are determined as:L _(θθ) U _(θ) +L _(θr) U _(r)+Ω₂ U _(θ)=0L _(rθ) U _(θ) +L _(rr) U _(r)+Ω² U _(r)=0Where L is an operator in terms of

${\nabla_{n}^{2} = \frac{d}{d\eta}}\left( {1 + \eta^{2}} \right)\frac{d}{d\eta}$η = cos  θ$\Omega = {\frac{\omega a}{c_{p}}\mspace{14mu}{dimensionless}\mspace{14mu}{frequency}\mspace{14mu}{of}\mspace{14mu}{shell}}$Where

c_(p) = phase  velocity  of  compressional  waves$c_{p} = \sqrt{\frac{E}{\rho\left( {1 - v^{2}} \right)}}$So

$\omega = {\frac{\Omega}{a}c_{p}}$ $f = {\frac{\Omega}{2\pi a}c_{p}}$${f = {\frac{\Omega}{2\pi a}\sqrt{\frac{E}{\rho\left( {1 - v^{2}} \right)}}}}{frequency}\mspace{14mu}{of}\mspace{14mu}{vibration}\mspace{14mu}{of}\mspace{14mu} a\mspace{14mu}{spherical}\mspace{14mu}{shell}$Free Vibration

We can represent

${{U_{r}(\eta)} = {\sum\limits_{n = 0}^{\infty}{U_{rn}{P_{n}(\eta)}e^{{- i}\omega t}}}}{{U_{\theta}(\eta)} = {\sum\limits_{n = 0}^{\infty}{U_{\theta n}\sqrt{\left( {1 - \eta^{2}} \right)}\left( {d{P_{n}/d}\eta} \right)e^{{- i}\omega t}}}}$

And therefore[Ω₂−(1+β²)(ν+λ_(n)−1)]u _(θn)−[β²(ν+λ_(n)−1)+(1+ν)]u_(rn)=0−λ_(n)[β²(ν+λ_(n)−1)+(1+ν)]u_(θn)+[Ω²−2(1+ν)−β²λ_(n)(ν+λ_(n)−1)]u _(rn)=0

For the above to have a non-trivial solution the following must besatisfied:Ω⁴−[1+3ν+λ_(n)−β²(1−ν−λ_(n) ²−νλ_(n))]β²+(λ_(n)−2)(1−ν²)+β²[λ_(n)³−4λ_(n) ²+λ_(n)(5−ν²)−2(1−ν²)]=0Where

λ_(n) = n(n + 1) $\beta = \frac{h^{2}}{12a^{2}}$Or[Ω²−Ω_(n) ⁽¹⁾ ² ][Ω²−Ω_(n) ⁽²⁾ ² ]=0

-   -   Ω_(n) ⁽¹⁾ lower natural frequencies    -   Ω_(n) ⁽²⁾ higher natural frequencies

From the modal characteristic equation of 3D shell theory:

$\Delta_{n} = {{\begin{matrix}{T_{11}^{(1)}\left( {\alpha r_{i}} \right)} & {T_{11}^{(2)}\left( {\alpha r_{i}} \right)} \\{T_{11}^{(1)}\left( {\alpha r_{0}} \right)} & {T_{11}^{(2)}\left( {\alpha r_{0}} \right)}\end{matrix}} = {{0\mspace{14mu}{for}\mspace{14mu} n} = 0}}$ThereforeT ₁₁ ⁽¹⁾(αr _(i))T ₁₁ ⁽²⁾(αr ₀)−T ₁₁ ⁽²⁾(αr _(i))T ₁₁ ⁽¹⁾(αr ₀)=0WhereT ₁₁ ^((i))(αr)=(n ² −n−½β² r ²)Z _(n) ^((i))(αr)+2αrZ _(n+1) ^((i))(αr)Where

$Z_{n}^{(1)} = {{j_{n}\left( {kr} \right)} = {\sqrt{\frac{\pi}{2{kr}}}{J_{n + \frac{1}{2}}\left( {kr} \right)}}}$$Z_{n}^{(2)} = {{y_{n}\left( {kr} \right)} = {\sqrt{\frac{\pi}{2{kr}}}{J_{n + \frac{1}{2}}\left( {kr} \right)}}}$$\alpha = \frac{\omega}{c_{1}}$ $\beta = \frac{\omega}{c_{2}}$$\Omega^{1} = {\frac{\omega}{c_{1}}a}$${T_{11}^{(1)}\left( {\alpha r_{i}} \right)} = {{\left( {n^{2} - n - {\frac{1}{2}\beta^{2}r_{i}^{2}}} \right){j_{n}\left( {kr_{i}} \right)}} + {2\alpha r_{i}{j_{n + 1}\left( {\alpha r_{i}} \right)}}}$${T_{11}^{(2)}\left( {\alpha r_{i}} \right)} = {{\left( {n^{2} - n - {\frac{1}{2}\beta^{2}r_{i}^{2}}} \right){y_{n}\left( {kr_{i}} \right)}} + {2\alpha r_{i}{y_{n + 1}\left( {\alpha r_{i}} \right)}}}$${T_{11}^{(1)}\left( {\alpha r_{0}} \right)} = {{\left( {n^{2} - n - {\frac{1}{2}\beta^{2}r_{0}^{2}}} \right){j_{n}\left( {kr_{0}} \right)}} + {2\alpha r_{0}{j_{n + 1}\left( {\alpha r_{0}} \right)}}}$${T_{11}^{(2)}\left( {\alpha r_{0}} \right)} = {{\left( {n^{2} - n - {\frac{1}{2}\beta^{2}r_{0}^{2}}} \right){y_{n}\left( {kr_{0}} \right)}} + {2\alpha r_{0}{y_{n + 1}\left( {\alpha r_{0}} \right)}}}$

It will be appreciated by those skilled in the art having the benefit ofthis disclosure that this a miniaturized device to sterilize fromCOVID-19 and other viruses and bacteria provides a an improved mannerfor sanitizing areas and clearing them of viruses and other biologicalmaterials. It should be understood that the drawings and detaileddescription herein are to be regarded in an illustrative rather than arestrictive manner and are not intended to be limiting to the particularforms and examples disclosed. On the contrary, included are any furthermodifications, changes, rearrangements, substitutions, alternatives,design choices, and embodiments apparent to those of ordinary skill inthe art, without departing from the spirit and scope hereof, as definedby the following claims. Thus, it is intended that the following claimsbe interpreted to embrace all such further modifications, changes,rearrangements, substitutions, alternatives, design choices, andembodiments.

What is claimed is:
 1. A system for sterilizing viruses, comprising:beam generation circuitry for generating a Hermite-Gaussian beam havingradiating energy therein at a predetermined frequency for generatingmechanical longitudinal eigen vibrations to a thin shell capsid of avirus to destroy the virus and a predetermined cartesianHermite-Gaussian intensity for imparting transverse shear forces to anicosahedral lattice structure of the virus to destroy the icosahedrallattice structure of the virus; a controller for controlling the beamgeneration circuitry to generate the Hermite-Gaussian beam at thepredetermined frequency for generating the mechanical longitudinal eigenvibrations to the thin shell capsid of the virus to destroy the virusand the predetermined cartesian Hermite-Gaussian intensity for impartingthe transverse shear forces to an icosahedral lattice structure of thevirus to destroy the icosahedral lattice structure of the virus, whereinthe predetermined frequency equals a resonance frequency of the virushaving the thin shell capsid and the Hermite-Gaussian beam is ordered asa structured Hermite-Gaussian beam with the predetermined cartesianHermite-Gaussian intensity of a specific order to destroy theicosahedral lattice structure of the virus, further wherein thepredetermined frequency is determined responsive to a plurality ofparameters from an influenza virus; wherein the predetermined frequencyinduces a mechanical resonance vibration at the resonance frequency ofthe virus for destroying a capsid of the virus; wherein theHermite-Gaussian beam at the predetermined frequency and thepredetermined cartesian Hermite-Gaussian intensity induces mechanicallongitudinal eigen-vibrations at the resonance frequency of the virushaving the thin shell capsid within the virus having the thin shellcapsid for destroying the virus having the thin shell capsid and impartsthe transverse shear forces to the icosahedral lattice structure of thevirus; and radiating circuitry for projecting a radiating wave on apredetermined location to destroy the virus having the thin shell capsidat the predetermined location.
 2. The system of claim 1, wherein thevirus comprises a Covid-19 virus.
 3. The system of claim 1, wherein thecontroller further determines a minimum magnitude of an electric fieldnecessary to induce the mechanical resonance vibration in the virus. 4.The system of claim 1, wherein the radiating circuitry comprises anantenna incorporated into a device from the group consisting of ahandheld flashlight, a light fixture, an electrical socket fixture and aceiling mounted fixture.
 5. The system of claim 1, wherein the beamgeneration circuitry generates a maximum transmit power required thatdoes not exceed a predetermined power density threshold.
 6. The systemof claim 1, wherein the beam generation circuitry further generates amicrowave beam and further, wherein the microwave beam generates aninactivation threshold within the virus responsive to microwave energywithin the microwave beam.
 7. The system of claim 1, wherein thecontroller causes generation of the radiating wave that has a structuredvector beam for imparting stresses and torsion to the virus to rupturethe capsid of the virus.
 8. A system for sterilizing viruses,comprising: beam generation circuitry for generating a Hermite-Gaussianbeam having radiating energy therein at a predetermined frequency forgenerating mechanical longitudinal eigen vibrations to a thin shellcapsid of a Covid-19 virus to destroy the Covid-19 virus; a controllerfor controlling the beam generation circuitry to generate theHermite-Gaussian beam at the predetermined frequency for generating themechanical longitudinal eigen vibrations to the thin shell capsid of theCovid-19 virus to destroy the Covid-19 virus, wherein the predeterminedfrequency equals a resonance frequency of the Covid-19 virus, furtherwherein the predetermined frequency is determined responsive to aplurality of parameters from an influenza virus; wherein thepredetermined frequency induces a mechanical resonance vibration at theresonance frequency of the Covid-19 virus for destroying the thin shellcapsid of the Covid-19 virus; wherein the Hermite-Gaussian beam at thepredetermined frequency induces mechanical longitudinal eigen-vibrationsat the resonance frequency of the virus having the thin shell capsidwithin the virus having the thin shell capsid for destroying the virushaving the thin shell capsid and imparts transverse shear forces toicosahedral lattice structure of the virus; and radiating circuitry forprojecting a radiating wave on a predetermined location to destroy theCovid-19 virus at the predetermined location.
 9. The system of claim 8,wherein the controller further determines a minimum magnitude of anelectric field necessary to induce the mechanical resonance vibration inthe Covid-19 virus.
 10. The system of claim 9, wherein the controllercauses the beam generation circuitry to generate the radiating wavehaving the minimum magnitude of the electric field.
 11. The system ofclaim 8, wherein the radiating circuitry comprises an antennaincorporated into a device from the group consisting of a handheldflashlight, a light fixture, an electrical socket fixture and a ceilingmounted fixture.
 12. The system of claim 8, wherein the beam generationcircuitry generates a maximum transmit power required that does notexceed a predetermined power density threshold.
 13. The system of claim8, wherein the beam generation circuitry further generates a microwavebeam and further wherein the microwave beam generates an inactivationthreshold within the Covid-19 virus responsive to microwave energywithin the microwave beam.
 14. The system of claim 8, wherein thecontroller causes generation of the radiating wave that has a structuredvector beam for imparting stresses and torsion to the Covid-19 virus torupture the thin shell capsid of the Covid-19 virus.